Vincent Selhorst-Jones

Limits at Infinity & Limits of Sequences

Slide Duration:

Section 1: Introduction
Introduction to Precalculus

10m 3s

Intro
0:00
Title of the Course
0:06
Different Names for the Course
0:07
Precalculus
0:12
Math Analysis
0:14
Trigonometry
0:16
Algebra III
0:20
Geometry II
0:24
College Algebra
0:30
Same Concepts
0:36
How do the Lessons Work?
0:54
Introducing Concepts
0:56
Apply Concepts
1:04
Go through Examples
1:25
Who is this Course For?
1:38
Those Who Need eExtra Help with Class Work
1:52
Those Working on Material but not in Formal Class at School
1:54
Those Who Want a Refresher
2:00
Try to Watch the Whole Lesson
2:20
Understanding is So Important
3:56
What to Watch First
5:26
Lesson #2: Sets, Elements, and Numbers
5:30
Lesson #7: Idea of a Function
5:33
Lesson #6: Word Problems
6:04
What to Watch First, cont.
6:46
Lesson #2: Sets, Elements and Numbers
6:56
Lesson #3: Variables, Equations, and Algebra
6:58
Lesson #4: Coordinate Systems
7:00
Lesson #5: Midpoint, Distance, the Pythagorean Theorem and Slope
7:02
Lesson #6: Word Problems
7:10
Lesson #7: Idea of a Function
7:12
Lesson #8: Graphs
7:14
Graphing Calculator Appendix
7:40
What to Watch Last
8:46
Let's get Started!
9:48
Sets, Elements, & Numbers

45m 11s

Intro
0:00
Introduction
0:05
Sets and Elements
1:19
Set
1:20
Element
1:23
Name a Set
2:20
Order The Elements Appear In Has No Effect on the Set
2:55
Describing/ Defining Sets
3:28
Directly Say All the Elements
3:36
Clearly Describing All the Members of the Set
3:55
Describing the Quality (or Qualities) Each member Of the Set Has In Common
4:32
Symbols: 'Element of' and 'Subset of'
6:01
Symbol is ∈
6:03
Subset Symbol is ⊂
6:35
Empty Set
8:07
Symbol is ∅
8:20
Since It's Empty, It is a Subset of All Sets
8:44
Union and Intersection
9:54
Union Symbol is ∪
10:08
Intersection Symbol is ∩
10:18
Sets Can Be Weird Stuff
12:26
Can Have Elements in a Set
12:50
We Can Have Infinite Sets
13:09
Example
13:22
Consider a Set Where We Take a Word and Then Repeat It An Ever Increasing Number of Times
14:08
This Set Has Infinitely Many Distinct Elements
14:40
Numbers as Sets
16:03
Natural Numbers ℕ
16:16
Including 0 and the Negatives ℤ
18:13
Rational Numbers ℚ
19:27
Can Express Rational Numbers with Decimal Expansions
22:05
Irrational Numbers
23:37
Real Numbers ℝ: Put the Rational and Irrational Numbers Together
25:15
Interval Notation and the Real Numbers
26:45
Include the End Numbers
27:06
Exclude the End Numbers
27:33
Example
28:28
Interval Notation: Infinity
29:09
Use -∞ or ∞ to Show an Interval Going on Forever in One Direction or the Other
29:14
Always Use Parentheses
29:50
Examples
30:27
Example 1
31:23
Example 2
35:26
Example 3
38:02
Example 4
42:21
Variables, Equations, & Algebra

35m 31s

Intro
0:00
What is a Variable?
0:05
A Variable is a Placeholder for a Number
0:11
Affects the Output of a Function or a Dependent Variable
0:24
Naming Variables
1:51
Useful to Use Symbols
2:21
What is a Constant?
4:14
A Constant is a Fixed, Unchanging Number
4:28
We Might Refer to a Symbol Representing a Number as a Constant
4:51
What is a Coefficient?
5:33
A Coefficient is a Multiplicative Factor on a Variable
5:37
Not All Coefficients are Constants
5:51
Expressions and Equations
6:42
An Expression is a String of Mathematical Symbols That Make Sense Used Together
7:05
An Equation is a Statement That Two Expression Have the Same Value
8:20
The Idea of Algebra
8:51
Equality
8:59
If Two Things Are the Same *Equal), Then We Can Do the Exact Same Operation to Both and the Results Will Be the Same
9:41
Always Do The Exact Same Thing to Both Sides
12:22
Solving Equations
13:23
When You Are Asked to Solve an Equation, You Are Being Asked to Solve for Something
13:33
Look For What Values Makes the Equation True
13:38
Isolate the Variable by Doing Algebra
14:37
Order of Operations
16:02
Why Certain Operations are Grouped
17:01
When You Don't Have to Worry About Order
17:39
Distributive Property
18:15
It Allows Multiplication to Act Over Addition in Parentheses
18:23
We Can Use the Distributive Property in Reverse to Combine Like Terms
19:05
Substitution
20:03
Use Information From One Equation in Another Equation
20:07
20:44
Example 1
23:17
Example 2
25:49
Example 3
28:11
Example 4
30:02
Coordinate Systems

35m 2s

Intro
0:00
Inherent Order in ℝ
0:05
Real Numbers Come with an Inherent Order
0:11
Positive Numbers
0:21
Negative Numbers
0:58
'Less Than' and 'Greater Than'
2:04
2:56
Inequality
4:06
Less Than or Equal and Greater Than or Equal
4:51
One Dimension: The Number Line
5:36
Graphically Represent ℝ on a Number Line
5:43
Note on Infinities
5:57
With the Number Line, We Can Directly See the Order We Put on ℝ
6:35
Ordered Pairs
7:22
Example
7:34
Allows Us to Talk About Two Numbers at the Same Time
9:41
Ordered Pairs of Real Numbers Cannot be Put Into an Order Like we Did with ℝ
10:41
Two Dimensions: The Plane
13:13
We Can Represent Ordered Pairs with the Plane
13:24
Intersection is known as the Origin
14:31
Plotting the Point
14:32
Plane = Coordinate Plane = Cartesian Plane = ℝ²
17:46
18:50
19:04
19:21
20:04
20:20
Three Dimensions: Space
21:02
Create Ordered Triplets
21:09
Visually Represent This
21:19
Three-Dimension = Space = ℝ³
21:47
Higher Dimensions
22:24
If We Have n Dimensions, We Call It n-Dimensional Space or ℝ to the nth Power
22:31
We Can Represent Places In This n-Dimensional Space As Ordered Groupings of n Numbers
22:41
Hard to Visualize Higher Dimensional Spaces
23:18
Example 1
25:07
Example 2
26:10
Example 3
28:58
Example 4
31:05
Midpoints, Distance, the Pythagorean Theorem, & Slope

48m 43s

Intro
0:00
Introduction
0:07
Midpoint: One Dimension
2:09
Example of Something More Complex
2:31
Use the Idea of a Middle
3:28
Find the Midpoint of Arbitrary Values a and b
4:17
How They're Equivalent
5:05
Official Midpoint Formula
5:46
Midpoint: Two Dimensions
6:19
The Midpoint Must Occur at the Horizontal Middle and the Vertical Middle
6:38
Arbitrary Pair of Points Example
7:25
Distance: One Dimension
9:26
Absolute Value
10:54
Idea of Forcing Positive
11:06
Distance: One Dimension, Formula
11:47
Distance Between Arbitrary a and b
11:48
Absolute Value Helps When the Distance is Negative
12:41
Distance Formula
12:58
The Pythagorean Theorem
13:24
a²+b²=c²
13:50
Distance: Two Dimensions
14:59
Break Into Horizontal and Vertical Parts and then Use the Pythagorean Theorem
15:16
Distance Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
16:21
Slope
19:30
Slope is the Rate of Change
19:41
m = rise over run
21:27
Slope Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
22:31
Interpreting Slope
24:12
Positive Slope and Negative Slope
25:40
m=1, m=0, m=-1
26:48
Example 1
28:25
Example 2
31:42
Example 3
36:40
Example 4
42:48
Word Problems

56m 31s

Intro
0:00
Introduction
0:05
What is a Word Problem?
0:45
Describes Any Problem That Primarily Gets Its Ideas Across With Words Instead of Math Symbols
0:48
Requires Us to Think
1:32
Why Are They So Hard?
2:11
Reason 1: No Simple Formula to Solve Them
2:16
Reason 2: Harder to Teach Word Problems
2:47
You Can Learn How to Do Them!
3:51
7:57
'But I'm Never Going to Use This In Real Life'
9:46
Solving Word Problems
12:58
First: Understand the Problem
13:37
Second: What Are You Looking For?
14:33
Third: Set Up Relationships
16:21
Fourth: Solve It!
17:48
Summary of Method
19:04
Examples on Things Other Than Math
20:21
Math-Specific Method: What You Need Now
25:30
Understand What the Problem is Talking About
25:37
Set Up and Name Any Variables You Need to Know
25:56
Set Up Equations Connecting Those Variables to the Information in the Problem Statement
26:02
Use the Equations to Solve for an Answer
26:14
Tip
26:58
Draw Pictures
27:22
Breaking Into Pieces
28:28
Try Out Hypothetical Numbers
29:52
Student Logic
31:27
Jump In!
32:40
Example 1
34:03
Example 2
39:15
Example 3
44:22
Example 4
50:24
Section 2: Functions
Idea of a Function

39m 54s

Intro
0:00
Introduction
0:04
What is a Function?
1:06
A Visual Example and Non-Example
1:30
Function Notation
3:47
f(x)
4:05
Express What Sets the Function Acts On
5:45
Metaphors for a Function
6:17
Transformation
6:28
Map
7:17
Machine
8:56
Same Input Always Gives Same Output
10:01
If We Put the Same Input Into a Function, It Will Always Produce the Same Output
10:11
Example of Something That is Not a Function
11:10
A Non-Numerical Example
12:10
The Functions We Will Use
15:05
Unless Told Otherwise, We Will Assume Every Function Takes in Real Numbers and Outputs Real Numbers
15:11
Usually Told the Rule of a Given Function
15:27
How To Use a Function
16:18
Apply the Rule to Whatever Our Input Value Is
16:28
Make Sure to Wrap Your Substitutions in Parentheses
17:09
Functions and Tables
17:36
Table of Values, Sometimes Called a T-Table
17:46
Example
17:56
Domain: What Goes In
18:55
The Domain is the Set of all Inputs That the Function Can Accept
18:56
Example
19:40
Range: What Comes Out
21:27
The Range is the Set of All Possible Outputs a Function Can Assign
21:34
Example
21:49
Another Example Would Be Our Initial Function From Earlier in This Lesson
22:29
Example 1
23:45
Example 2
25:22
Example 3
27:27
Example 4
29:23
Example 5
33:33
Graphs

58m 26s

Intro
0:00
Introduction
0:04
How to Interpret Graphs
1:17
Input / Independent Variable
1:47
Output / Dependent Variable
2:00
Graph as Input ⇒ Output
2:23
One Way to Think of a Graph: See What Happened to Various Inputs
2:25
Example
2:47
Graph as Location of Solution
4:20
A Way to See Solutions
4:36
Example
5:20
Which Way Should We Interpret?
7:13
Easiest to Think In Terms of How Inputs Are Mapped to Outputs
7:20
Sometimes It's Easier to Think In Terms of Solutions
8:39
Pay Attention to Axes
9:50
Axes Tell Where the Graph Is and What Scale It Has
10:09
Often, The Axes Will Be Square
10:14
Example
12:06
Arrows or No Arrows?
16:07
Will Not Use Arrows at the End of Our Graphs
17:13
Graph Stops Because It Hits the Edge of the Graphing Axes, Not Because the Function Stops
17:18
How to Graph
19:47
Plot Points
20:07
Connect with Curves
21:09
If You Connect with Straight Lines
21:44
Graphs of Functions are Smooth
22:21
More Points ⇒ More Accurate
23:38
Vertical Line Test
27:44
If a Vertical Line Could Intersect More Than One Point On a Graph, It Can Not Be the Graph of a Function
28:41
Every Point on a Graph Tells Us Where the x-Value Below is Mapped
30:07
Domain in Graphs
31:37
The Domain is the Set of All Inputs That a Function Can Accept
31:44
Be Aware That Our Function Probably Continues Past the Edge of Our 'Viewing Window'
33:19
Range in Graphs
33:53
Graphing Calculators: Check the Appendix!
36:55
Example 1
38:37
Example 2
45:19
Example 3
50:41
Example 4
53:28
Example 5
55:50
Properties of Functions

48m 49s

Intro
0:00
Introduction
0:05
Increasing Decreasing Constant
0:43
Looking at a Specific Graph
1:15
Increasing Interval
2:39
Constant Function
4:15
Decreasing Interval
5:10
Find Intervals by Looking at the Graph
5:32
Intervals Show x-values; Write in Parentheses
6:39
Maximum and Minimums
8:48
Relative (Local) Max/Min
10:20
Formal Definition of Relative Maximum
12:44
Formal Definition of Relative Minimum
13:05
Max/Min, More Terms
14:18
Definition of Extrema
15:01
Average Rate of Change
16:11
Drawing a Line for the Average Rate
16:48
Using the Slope of the Secant Line
17:36
Slope in Function Notation
18:45
Zeros/Roots/x-intercepts
19:45
What Zeros in a Function Mean
20:25
Even Functions
22:30
Odd Functions
24:36
Even/Odd Functions and Graphs
26:28
Example of an Even Function
27:12
Example of an Odd Function
28:03
Example 1
29:35
Example 2
33:07
Example 3
40:32
Example 4
42:34
Function Petting Zoo

29m 20s

Intro
0:00
Introduction
0:04
Don't Forget that Axes Matter!
1:44
The Constant Function
2:40
The Identity Function
3:44
The Square Function
4:40
The Cube Function
5:44
The Square Root Function
6:51
The Reciprocal Function
8:11
The Absolute Value Function
10:19
The Trigonometric Functions
11:56
f(x)=sin(x)
12:12
f(x)=cos(x)
12:24
Alternate Axes
12:40
The Exponential and Logarithmic Functions
13:35
Exponential Functions
13:44
Logarithmic Functions
14:24
Alternating Axes
15:17
Transformations and Compositions
16:08
Example 1
17:52
Example 2
18:33
Example 3
20:24
Example 4
26:07
Transformation of Functions

48m 35s

Intro
0:00
Introduction
0:04
Vertical Shift
1:12
Graphical Example
1:21
A Further Explanation
2:16
Vertical Stretch/Shrink
3:34
Graph Shrinks
3:46
Graph Stretches
3:51
A Further Explanation
5:07
Horizontal Shift
6:49
Moving the Graph to the Right
7:28
Moving the Graph to the Left
8:12
A Further Explanation
8:19
Understanding Movement on the x-axis
8:38
Horizontal Stretch/Shrink
12:59
Shrinking the Graph
13:40
Stretching the Graph
13:48
A Further Explanation
13:55
Understanding Stretches from the x-axis
14:12
Vertical Flip (aka Mirror)
16:55
Example Graph
17:07
Multiplying the Vertical Component by -1
17:18
Horizontal Flip (aka Mirror)
18:43
Example Graph
19:01
Multiplying the Horizontal Component by -1
19:54
Summary of Transformations
22:11
Stacking Transformations
24:46
Order Matters
25:20
Transformation Example
25:52
Example 1
29:21
Example 2
34:44
Example 3
38:10
Example 4
43:46
Composite Functions

33m 24s

Intro
0:00
Introduction
0:04
Arithmetic Combinations
0:40
Basic Operations
1:20
Definition of the Four Arithmetic Combinations
1:40
Composite Functions
2:53
The Function as a Machine
3:32
Function Compositions as Multiple Machines
3:59
Notation for Composite Functions
4:46
Two Formats
6:02
Another Visual Interpretation
7:17
How to Use Composite Functions
8:21
Example of on Function acting on Another
9:17
Example 1
11:03
Example 2
15:27
Example 3
21:11
Example 4
27:06
Piecewise Functions

51m 42s

Intro
0:00
Introduction
0:04
Analogies to a Piecewise Function
1:16
Different Potatoes
1:41
Factory Production
2:27
Notations for Piecewise Functions
3:39
Notation Examples from Analogies
6:11
Example of a Piecewise (with Table)
7:24
Example of a Non-Numerical Piecewise
11:35
Graphing Piecewise Functions
14:15
Graphing Piecewise Functions, Example
16:26
Continuous Functions
16:57
Statements of Continuity
19:30
Example of Continuous and Non-Continuous Graphs
20:05
Interesting Functions: the Step Function
22:00
Notation for the Step Function
22:40
How the Step Function Works
22:56
Graph of the Step Function
25:30
Example 1
26:22
Example 2
28:49
Example 3
36:50
Example 4
46:11
Inverse Functions

49m 37s

Intro
0:00
Introduction
0:04
Analogy by picture
1:10
How to Denote the inverse
1:40
What Comes out of the Inverse
1:52
Requirement for Reversing
2:02
2:12
The Importance of Information
2:45
One-to-One
4:04
Requirement for Reversibility
4:21
When a Function has an Inverse
4:43
One-to-One
5:13
Not One-to-One
5:50
Not a Function
6:19
Horizontal Line Test
7:01
How to the test Works
7:12
One-to-One
8:12
Not One-to-One
8:45
Definition: Inverse Function
9:12
Formal Definition
9:21
Caution to Students
10:02
Domain and Range
11:12
Finding the Range of the Function Inverse
11:56
Finding the Domain of the Function Inverse
12:11
Inverse of an Inverse
13:09
Its just x!
13:26
Proof
14:03
Graphical Interpretation
17:07
Horizontal Line Test
17:20
Graph of the Inverse
18:04
Swapping Inputs and Outputs to Draw Inverses
19:02
How to Find the Inverse
21:03
What We Are Looking For
21:21
Reversing the Function
21:38
A Method to Find Inverses
22:33
Check Function is One-to-One
23:04
Swap f(x) for y
23:25
Interchange x and y
23:41
Solve for y
24:12
Replace y with the inverse
24:40
25:01
Keeping Step 2 and 3 Straight
25:44
Switching to Inverse
26:12
Checking Inverses
28:52
How to Check an Inverse
29:06
Quick Example of How to Check
29:56
Example 1
31:48
Example 2
34:56
Example 3
39:29
Example 4
46:19
Variation Direct and Inverse

28m 49s

Intro
0:00
Introduction
0:06
Direct Variation
1:14
Same Direction
1:21
Common Example: Groceries
1:56
Different Ways to Say that Two Things Vary Directly
2:28
Basic Equation for Direct Variation
2:55
Inverse Variation
3:40
Opposite Direction
3:50
Common Example: Gravity
4:53
Different Ways to Say that Two Things Vary Indirectly
5:48
Basic Equation for Indirect Variation
6:33
Joint Variation
7:27
Equation for Joint Variation
7:53
Explanation of the Constant
8:48
Combined Variation
9:35
Gas Law as a Combination
9:44
Single Constant
10:33
Example 1
10:49
Example 2
13:34
Example 3
15:39
Example 4
19:48
Section 3: Polynomials
Intro to Polynomials

38m 41s

Intro
0:00
Introduction
0:04
Definition of a Polynomial
1:04
Starting Integer
2:06
Structure of a Polynomial
2:49
The a Constants
3:34
Polynomial Function
5:13
Polynomial Equation
5:23
Polynomials with Different Variables
5:36
Degree
6:23
Informal Definition
6:31
Find the Largest Exponent Variable
6:44
Quick Examples
7:36
Special Names for Polynomials
8:59
Based on the Degree
9:23
Based on the Number of Terms
10:12
Distributive Property (aka 'FOIL')
11:37
Basic Distributive Property
12:21
Distributing Two Binomials
12:55
Longer Parentheses
15:12
Reverse: Factoring
17:26
Long-Term Behavior of Polynomials
17:48
Examples
18:13
Controlling Term--Term with the Largest Exponent
19:33
Positive and Negative Coefficients on the Controlling Term
20:21
22:07
Even Degree, Positive Coefficient
22:13
Even Degree, Negative Coefficient
22:39
Odd Degree, Positive Coefficient
23:09
Odd Degree, Negative Coefficient
23:27
Example 1
25:11
Example 2
27:16
Example 3
31:16
Example 4
34:41
Roots (Zeros) of Polynomials

41m 7s

Intro
0:00
Introduction
0:05
Roots in Graphs
1:17
The x-intercepts
1:33
How to Remember What 'Roots' Are
1:50
Naïve Attempts
2:31
Isolating Variables
2:45
Failures of Isolating Variables
3:30
Missing Solutions
4:59
Factoring: How to Find Roots
6:28
How Factoring Works
6:36
Why Factoring Works
7:20
Steps to Finding Polynomial Roots
9:21
Factoring: How to Find Roots CAUTION
10:08
Factoring is Not Easy
11:32
13:08
13:21
Form of Factored Binomials
13:38
Factoring Examples
14:40
16:58
Factoring Higher Degree Polynomials
18:19
Factoring a Cubic
18:32
19:04
Factoring: Roots Imply Factors
19:54
Where a Root is, A Factor Is
20:01
How to Use Known Roots to Make Factoring Easier
20:35
Not all Polynomials Can be Factored
22:30
Irreducible Polynomials
23:27
Complex Numbers Help
23:55
Max Number of Roots/Factors
24:57
Limit to Number of Roots Equal to the Degree
25:18
Why there is a Limit
25:25
Max Number of Peaks/Valleys
26:39
Shape Information from Degree
26:46
Example Graph
26:54
Max, But Not Required
28:00
Example 1
28:37
Example 2
31:21
Example 3
36:12
Example 4
38:40
Completing the Square and the Quadratic Formula

39m 43s

Intro
0:00
Introduction
0:05
Square Roots and Equations
0:51
Taking the Square Root to Find the Value of x
0:55
Getting the Positive and Negative Answers
1:05
Completing the Square: Motivation
2:04
Polynomials that are Easy to Solve
2:20
Making Complex Polynomials Easy to Solve
3:03
Steps to Completing the Square
4:30
Completing the Square: Method
7:22
Move C over
7:35
Divide by A
7:44
Find r
7:59
Add to Both Sides to Complete the Square
8:49
9:56
11:38
Derivation
11:43
Final Form
12:23
13:38
How Many Roots?
14:53
The Discriminant
15:47
What the Discriminant Tells Us: How Many Roots
15:58
How the Discriminant Works
16:30
Example 1: Complete the Square
18:24
22:00
Example 3: Solve for Zeroes
25:28
Example 4: Using the Quadratic Formula
30:52

45m 34s

Intro
0:00
Introduction
0:05
Parabolas
0:35
Examples of Different Parabolas
1:06
Axis of Symmetry and Vertex
1:28
Drawing an Axis of Symmetry
1:51
Placing the Vertex
2:28
Looking at the Axis of Symmetry and Vertex for other Parabolas
3:09
Transformations
4:18
Reviewing Transformation Rules
6:28
Note the Different Horizontal Shift Form
7:45
8:54
The Constants: k, h, a
9:05
Transformations Formed
10:01
Analyzing Different Parabolas
10:10
Switching Forms by Completing the Square
11:43
Vertex of a Parabola
16:30
Vertex at (h, k)
16:47
Vertex in Terms of a, b, and c Coefficients
17:28
Minimum/Maximum at Vertex
18:19
When a is Positive
18:25
When a is Negative
18:52
Axis of Symmetry
19:54
Incredibly Minor Note on Grammar
20:52
Example 1
21:48
Example 2
26:35
Example 3
28:55
Example 4
31:40
Intermediate Value Theorem and Polynomial Division

46m 8s

Intro
0:00
Introduction
0:05
Reminder: Roots Imply Factors
1:32
The Intermediate Value Theorem
3:41
The Basis: U between a and b
4:11
U is on the Function
4:52
Intermediate Value Theorem, Proof Sketch
5:51
If Not True, the Graph Would Have to Jump
5:58
But Graph is Defined as Continuous
6:43
Finding Roots with the Intermediate Value Theorem
7:01
Picking a and b to be of Different Signs
7:10
Must Be at Least One Root
7:46
Dividing a Polynomial
8:16
Using Roots and Division to Factor
8:38
Long Division Refresher
9:08
The Division Algorithm
12:18
How It Works to Divide Polynomials
12:37
The Parts of the Equation
13:24
Rewriting the Equation
14:47
Polynomial Long Division
16:20
Polynomial Long Division In Action
16:29
One Step at a Time
20:51
Synthetic Division
22:46
Setup
23:11
Synthetic Division, Example
24:44
Which Method Should We Use
26:39
26:49
27:13
Example 1
29:24
Example 2
31:27
Example 3
36:22
Example 4
40:55
Complex Numbers

45m 36s

Intro
0:00
Introduction
0:04
A Wacky Idea
1:02
The Definition of the Imaginary Number
1:22
How it Helps Solve Equations
2:20
Square Roots and Imaginary Numbers
3:15
Complex Numbers
5:00
Real Part and Imaginary Part
5:20
When Two Complex Numbers are Equal
6:10
6:40
Deal with Real and Imaginary Parts Separately
7:36
Two Quick Examples
7:54
Multiplication
9:07
FOIL Expansion
9:14
Note What Happens to the Square of the Imaginary Number
9:41
Two Quick Examples
10:22
Division
11:27
Complex Conjugates
13:37
Getting Rid of i
14:08
How to Denote the Conjugate
14:48
Division through Complex Conjugates
16:11
Multiply by the Conjugate of the Denominator
16:28
Example
17:46
19:24
20:12
Conjugate Pairs
20:37
But Are the Complex Numbers 'Real'?
21:27
What Makes a Number Legitimate
25:38
Where Complex Numbers are Used
27:20
Still, We Won't See Much of C
29:05
Example 1
30:30
Example 2
33:15
Example 3
38:12
Example 4
42:07
Fundamental Theorem of Algebra

19m 9s

Intro
0:00
Introduction
0:05
Idea: Hidden Roots
1:16
Roots in Complex Form
1:42
All Polynomials Have Roots
2:08
Fundamental Theorem of Algebra
2:21
Where Are All the Imaginary Roots, Then?
3:17
All Roots are Complex
3:45
Real Numbers are a Subset of Complex Numbers
3:59
The n Roots Theorem
5:01
For Any Polynomial, Its Degree is Equal to the Number of Roots
5:11
Equivalent Statement
5:24
6:29
Non-Distinct Roots
6:59
Denoting Multiplicity
7:20
7:41
8:55
9:59
Proof Sketch of n Roots Theorem
10:45
First Root
11:36
Second Root
13:23
Continuation to Find all Roots
16:00
Section 4: Rational Functions
Rational Functions and Vertical Asymptotes

33m 22s

Intro
0:00
Introduction
0:05
Definition of a Rational Function
1:20
Examples of Rational Functions
2:30
Why They are Called 'Rational'
2:47
Domain of a Rational Function
3:15
Undefined at Denominator Zeros
3:25
Otherwise all Reals
4:16
Investigating a Fundamental Function
4:50
The Domain of the Function
5:04
What Occurs at the Zeroes of the Denominator
5:20
Idea of a Vertical Asymptote
6:23
What's Going On?
6:58
Approaching x=0 from the left
7:32
Approaching x=0 from the right
8:34
Dividing by Very Small Numbers Results in Very Large Numbers
9:31
Definition of a Vertical Asymptote
10:05
Vertical Asymptotes and Graphs
11:15
Drawing Asymptotes by Using a Dashed Line
11:27
The Graph Can Never Touch Its Undefined Point
12:00
Not All Zeros Give Asymptotes
13:02
Special Cases: When Numerator and Denominator Go to Zero at the Same Time
14:58
Cancel out Common Factors
15:49
How to Find Vertical Asymptotes
16:10
Figure out What Values Are Not in the Domain of x
16:24
Determine if the Numerator and Denominator Share Common Factors and Cancel
16:45
Find Denominator Roots
17:33
Note if Asymptote Approaches Negative or Positive Infinity
18:06
Example 1
18:57
Example 2
21:26
Example 3
23:04
Example 4
30:01
Horizontal Asymptotes

34m 16s

Intro
0:00
Introduction
0:05
Investigating a Fundamental Function
0:53
What Happens as x Grows Large
1:00
Different View
1:12
Idea of a Horizontal Asymptote
1:36
What's Going On?
2:24
What Happens as x Grows to a Large Negative Number
2:49
What Happens as x Grows to a Large Number
3:30
Dividing by Very Large Numbers Results in Very Small Numbers
3:52
Example Function
4:41
Definition of a Vertical Asymptote
8:09
Expanding the Idea
9:03
What's Going On?
9:48
What Happens to the Function in the Long Run?
9:51
Rewriting the Function
10:13
Definition of a Slant Asymptote
12:09
Symbolical Definition
12:30
Informal Definition
12:45
Beyond Slant Asymptotes
13:03
Not Going Beyond Slant Asymptotes
14:39
Horizontal/Slant Asymptotes and Graphs
15:43
How to Find Horizontal and Slant Asymptotes
16:52
How to Find Horizontal Asymptotes
17:12
Expand the Given Polynomials
17:18
Compare the Degrees of the Numerator and Denominator
17:40
How to Find Slant Asymptotes
20:05
Slant Asymptotes Exist When n+m=1
20:08
Use Polynomial Division
20:24
Example 1
24:32
Example 2
25:53
Example 3
26:55
Example 4
29:22
Graphing Asymptotes in a Nutshell

49m 7s

Intro
0:00
Introduction
0:05
A Process for Graphing
1:22
1. Factor Numerator and Denominator
1:50
2. Find Domain
2:53
3. Simplifying the Function
3:59
4. Find Vertical Asymptotes
4:59
5. Find Horizontal/Slant Asymptotes
5:24
6. Find Intercepts
7:35
7. Draw Graph (Find Points as Necessary)
9:21
Draw Graph Example
11:21
Vertical Asymptote
11:41
Horizontal Asymptote
11:50
Other Graphing
12:16
Test Intervals
15:08
Example 1
17:57
Example 2
23:01
Example 3
29:02
Example 4
33:37
Partial Fractions

44m 56s

Intro
0:00
Introduction: Idea
0:04
Introduction: Prerequisites and Uses
1:57
Proper vs. Improper Polynomial Fractions
3:11
Possible Things in the Denominator
4:38
Linear Factors
6:16
Example of Linear Factors
7:03
Multiple Linear Factors
7:48
8:25
9:26
9:49
Mixing Factor Types
10:28
Figuring Out the Numerator
11:10
How to Solve for the Constants
11:30
Quick Example
11:40
Example 1
14:29
Example 2
18:35
Example 3
20:33
Example 4
28:51
Section 5: Exponential & Logarithmic Functions
Understanding Exponents

35m 17s

Intro
0:00
Introduction
0:05
Fundamental Idea
1:46
Expanding the Idea
2:28
Multiplication of the Same Base
2:40
Exponents acting on Exponents
3:45
Different Bases with the Same Exponent
4:31
To the Zero
5:35
To the First
5:45
Fundamental Rule with the Zero Power
6:35
To the Negative
7:45
Any Number to a Negative Power
8:14
A Fraction to a Negative Power
9:58
Division with Exponential Terms
10:41
To the Fraction
11:33
Square Root
11:58
Any Root
12:59
Summary of Rules
14:38
To the Irrational
17:21
Example 1
20:34
Example 2
23:42
Example 3
27:44
Example 4
31:44
Example 5
33:15
Exponential Functions

47m 4s

Intro
0:00
Introduction
0:05
Definition of an Exponential Function
0:48
Definition of the Base
1:02
Restrictions on the Base
1:16
Computing Exponential Functions
2:29
Harder Computations
3:10
When to Use a Calculator
3:21
Graphing Exponential Functions: a>1
6:02
Three Examples
6:13
What to Notice on the Graph
7:44
A Story
8:27
Story Diagram
9:15
Increasing Exponentials
11:29
Story Morals
14:40
Application: Compound Interest
15:15
Compounding Year after Year
16:01
Function for Compounding Interest
16:51
A Special Number: e
20:55
Expression for e
21:28
Where e stabilizes
21:55
Application: Continuously Compounded Interest
24:07
Equation for Continuous Compounding
24:22
Exponential Decay 0<a<1
25:50
Three Examples
26:11
Why they 'lose' value
26:54
Example 1
27:47
Example 2
33:11
Example 3
36:34
Example 4
41:28
Introduction to Logarithms

40m 31s

Intro
0:00
Introduction
0:04
Definition of a Logarithm, Base 2
0:51
Log 2 Defined
0:55
Examples
2:28
Definition of a Logarithm, General
3:23
Examples of Logarithms
5:15
Problems with Unusual Bases
7:38
Shorthand Notation: ln and log
9:44
base e as ln
10:01
base 10 as log
10:34
Calculating Logarithms
11:01
using a calculator
11:34
issues with other bases
11:58
Graphs of Logarithms
13:21
Three Examples
13:29
Slow Growth
15:19
Logarithms as Inverse of Exponentiation
16:02
Using Base 2
16:05
General Case
17:10
Looking More Closely at Logarithm Graphs
19:16
The Domain of Logarithms
20:41
21:08
The Alternate
24:00
Example 1
25:59
Example 2
30:03
Example 3
32:49
Example 4
37:34
Properties of Logarithms

42m 33s

Intro
0:00
Introduction
0:04
Basic Properties
1:12
Inverse--log(exp)
1:43
A Key Idea
2:44
What We Get through Exponentiation
3:18
B Always Exists
4:50
Inverse--exp(log)
5:53
Logarithm of a Power
7:44
Logarithm of a Product
10:07
Logarithm of a Quotient
13:48
Caution! There Is No Rule for loga(M+N)
16:12
Summary of Properties
17:42
Change of Base--Motivation
20:17
No Calculator Button
20:59
A Specific Example
21:45
Simplifying
23:45
Change of Base--Formula
24:14
Example 1
25:47
Example 2
29:08
Example 3
31:14
Example 4
34:13
Solving Exponential and Logarithmic Equations

34m 10s

Intro
0:00
Introduction
0:05
One to One Property
1:09
Exponential
1:26
Logarithmic
1:44
Specific Considerations
2:02
One-to-One Property
3:30
Solving by One-to-One
4:11
Inverse Property
6:09
Solving by Inverses
7:25
Dealing with Equations
7:50
Example of Taking an Exponent or Logarithm of an Equation
9:07
A Useful Property
11:57
Bring Down Exponents
12:01
Try to Simplify
13:20
Extraneous Solutions
13:45
Example 1
16:37
Example 2
19:39
Example 3
21:37
Example 4
26:45
Example 5
29:37
Application of Exponential and Logarithmic Functions

48m 46s

Intro
0:00
Introduction
0:06
Applications of Exponential Functions
1:07
A Secret!
2:17
Natural Exponential Growth Model
3:07
Figure out r
3:34
A Secret!--Why Does It Work?
4:44
e to the r Morphs
4:57
Example
5:06
Applications of Logarithmic Functions
8:32
Examples
8:43
What Logarithms are Useful For
9:53
Example 1
11:29
Example 2
15:30
Example 3
26:22
Example 4
32:05
Example 5
39:19
Section 6: Trigonometric Functions
Angles

39m 5s

Intro
0:00
Degrees
0:22
Circle is 360 Degrees
0:48
Splitting a Circle
1:13
2:08
2:31
2:52
Half-Circle and Right Angle
4:00
6:24
6:52
Coterminal, Complementary, Supplementary Angles
7:23
Coterminal Angles
7:30
Complementary Angles
9:40
Supplementary Angles
10:08
Example 1: Dividing a Circle
10:38
Example 2: Converting Between Degrees and Radians
11:56
Example 3: Quadrants and Coterminal Angles
14:18
Extra Example 1: Common Angle Conversions
-1
Extra Example 2: Quadrants and Coterminal Angles
-2
Sine and Cosine Functions

43m 16s

Intro
0:00
Sine and Cosine
0:15
Unit Circle
0:22
Coordinates on Unit Circle
1:03
Right Triangles
1:52
2:25
Master Right Triangle Formula: SOHCAHTOA
2:48
Odd Functions, Even Functions
4:40
Example: Odd Function
4:56
Example: Even Function
7:30
Example 1: Sine and Cosine
10:27
Example 2: Graphing Sine and Cosine Functions
14:39
Example 3: Right Triangle
21:40
Example 4: Odd, Even, or Neither
26:01
Extra Example 1: Right Triangle
-1
Extra Example 2: Graphing Sine and Cosine Functions
-2
Sine and Cosine Values of Special Angles

33m 5s

Intro
0:00
45-45-90 Triangle and 30-60-90 Triangle
0:08
45-45-90 Triangle
0:21
30-60-90 Triangle
2:06
Mnemonic: All Students Take Calculus (ASTC)
5:21
Using the Unit Circle
5:59
New Angles
6:21
9:43
Mnemonic: All Students Take Calculus
10:13
13:11
16:48
Example 3: All Angles and Quadrants
20:21
Extra Example 1: Convert, Quadrant, Sine/Cosine
-1
Extra Example 2: All Angles and Quadrants
-2
Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D

52m 3s

Intro
0:00
Amplitude and Period of a Sine Wave
0:38
Sine Wave Graph
0:58
Amplitude: Distance from Middle to Peak
1:18
Peak: Distance from Peak to Peak
2:41
Phase Shift and Vertical Shift
4:13
Phase Shift: Distance Shifted Horizontally
4:16
Vertical Shift: Distance Shifted Vertically
6:48
Example 1: Amplitude/Period/Phase and Vertical Shift
8:04
Example 2: Amplitude/Period/Phase and Vertical Shift
17:39
Example 3: Find Sine Wave Given Attributes
25:23
Extra Example 1: Amplitude/Period/Phase and Vertical Shift
-1
Extra Example 2: Find Cosine Wave Given Attributes
-2
Tangent and Cotangent Functions

36m 4s

Intro
0:00
Tangent and Cotangent Definitions
0:21
Tangent Definition
0:25
Cotangent Definition
0:47
Master Formula: SOHCAHTOA
1:01
Mnemonic
1:16
Tangent and Cotangent Values
2:29
Remember Common Values of Sine and Cosine
2:46
90 Degrees Undefined
4:36
Slope and Menmonic: ASTC
5:47
Uses of Tangent
5:54
Example: Tangent of Angle is Slope
6:09
7:49
Example 1: Graph Tangent and Cotangent Functions
10:42
Example 2: Tangent and Cotangent of Angles
16:09
Example 3: Odd, Even, or Neither
18:56
Extra Example 1: Tangent and Cotangent of Angles
-1
Extra Example 2: Tangent and Cotangent of Angles
-2
Secant and Cosecant Functions

27m 18s

Intro
0:00
Secant and Cosecant Definitions
0:17
Secant Definition
0:18
Cosecant Definition
0:33
Example 1: Graph Secant Function
0:48
Example 2: Values of Secant and Cosecant
6:49
Example 3: Odd, Even, or Neither
12:49
Extra Example 1: Graph of Cosecant Function
-1
Extra Example 2: Values of Secant and Cosecant
-2
Inverse Trigonometric Functions

32m 58s

Intro
0:00
Arcsine Function
0:24
Restrictions between -1 and 1
0:43
Arcsine Notation
1:26
Arccosine Function
3:07
Restrictions between -1 and 1
3:36
Cosine Notation
3:53
Arctangent Function
4:30
Between -Pi/2 and Pi/2
4:44
Tangent Notation
5:02
Example 1: Domain/Range/Graph of Arcsine
5:45
Example 2: Arcsin/Arccos/Arctan Values
10:46
Example 3: Domain/Range/Graph of Arctangent
17:14
Extra Example 1: Domain/Range/Graph of Arccosine
-1
Extra Example 2: Arcsin/Arccos/Arctan Values
-2
Computations of Inverse Trigonometric Functions

31m 8s

Intro
0:00
Inverse Trigonometric Function Domains and Ranges
0:31
Arcsine
0:41
Arccosine
1:14
Arctangent
1:41
Example 1: Arcsines of Common Values
2:44
Example 2: Odd, Even, or Neither
5:57
Example 3: Arccosines of Common Values
12:24
Extra Example 1: Arctangents of Common Values
-1
Extra Example 2: Arcsin/Arccos/Arctan Values
-2
Section 7: Trigonometric Identities
Pythagorean Identity

19m 11s

Intro
0:00
Pythagorean Identity
0:17
Pythagorean Triangle
0:27
Pythagorean Identity
0:45
Example 1: Use Pythagorean Theorem to Prove Pythagorean Identity
1:14
Example 2: Find Angle Given Cosine and Quadrant
4:18
Example 3: Verify Trigonometric Identity
8:00
Extra Example 1: Use Pythagorean Identity to Prove Pythagorean Theorem
-1
Extra Example 2: Find Angle Given Cosine and Quadrant
-2
Identity Tan(squared)x+1=Sec(squared)x

23m 16s

Intro
0:00
Main Formulas
0:19
Companion to Pythagorean Identity
0:27
For Cotangents and Cosecants
0:52
How to Remember
0:58
Example 1: Prove the Identity
1:40
Example 2: Given Tan Find Sec
3:42
Example 3: Prove the Identity
7:45
Extra Example 1: Prove the Identity
-1
Extra Example 2: Given Sec Find Tan
-2

52m 52s

Intro
0:00
0:09
How to Remember
0:48
Cofunction Identities
1:31
How to Remember Graphically
1:44
Where to Use Cofunction Identities
2:52
Example 1: Derive the Formula for cos(A-B)
3:08
Example 2: Use Addition and Subtraction Formulas
16:03
Example 3: Use Addition and Subtraction Formulas to Prove Identity
25:11
Extra Example 1: Use cos(A-B) and Cofunction Identities
-1
Extra Example 2: Convert to Radians and use Formulas
-2
Double Angle Formulas

29m 5s

Intro
0:00
Main Formula
0:07
How to Remember from Addition Formula
0:18
Two Other Forms
1:35
Example 1: Find Sine and Cosine of Angle using Double Angle
3:16
Example 2: Prove Trigonometric Identity using Double Angle
9:37
Example 3: Use Addition and Subtraction Formulas
12:38
Extra Example 1: Find Sine and Cosine of Angle using Double Angle
-1
Extra Example 2: Prove Trigonometric Identity using Double Angle
-2
Half-Angle Formulas

43m 55s

Intro
0:00
Main Formulas
0:09
Confusing Part
0:34
Example 1: Find Sine and Cosine of Angle using Half-Angle
0:54
Example 2: Prove Trigonometric Identity using Half-Angle
11:51
Example 3: Prove the Half-Angle Formula for Tangents
18:39
Extra Example 1: Find Sine and Cosine of Angle using Half-Angle
-1
Extra Example 2: Prove Trigonometric Identity using Half-Angle
-2
Section 8: Applications of Trigonometry
Trigonometry in Right Angles

25m 43s

Intro
0:00
Master Formula for Right Angles
0:11
SOHCAHTOA
0:15
Only for Right Triangles
1:26
Example 1: Find All Angles in a Triangle
2:19
Example 2: Find Lengths of All Sides of Triangle
7:39
Example 3: Find All Angles in a Triangle
11:00
Extra Example 1: Find All Angles in a Triangle
-1
Extra Example 2: Find Lengths of All Sides of Triangle
-2
Law of Sines

56m 40s

Intro
0:00
Law of Sines Formula
0:18
SOHCAHTOA
0:27
Any Triangle
0:59
Graphical Representation
1:25
Solving Triangle Completely
2:37
When to Use Law of Sines
2:55
ASA, SAA, SSA, AAA
2:59
SAS, SSS for Law of Cosines
7:11
Example 1: How Many Triangles Satisfy Conditions, Solve Completely
8:44
Example 2: How Many Triangles Satisfy Conditions, Solve Completely
15:30
Example 3: How Many Triangles Satisfy Conditions, Solve Completely
28:32
Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely
-1
Extra Example 2: How Many Triangles Satisfy Conditions, Solve Completely
-2
Law of Cosines

49m 5s

Intro
0:00
Law of Cosines Formula
0:23
Graphical Representation
0:34
Relates Sides to Angles
1:00
Any Triangle
1:20
Generalization of Pythagorean Theorem
1:32
When to Use Law of Cosines
2:26
SAS, SSS
2:30
Heron's Formula
4:49
Semiperimeter S
5:11
Example 1: How Many Triangles Satisfy Conditions, Solve Completely
5:53
Example 2: How Many Triangles Satisfy Conditions, Solve Completely
15:19
Example 3: Find Area of a Triangle Given All Side Lengths
26:33
Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely
-1
Extra Example 2: Length of Third Side and Area of Triangle
-2
Finding the Area of a Triangle

27m 37s

Intro
0:00
Master Right Triangle Formula and Law of Cosines
0:19
SOHCAHTOA
0:27
Law of Cosines
1:23
Heron's Formula
2:22
Semiperimeter S
2:37
Example 1: Area of Triangle with Two Sides and One Angle
3:12
Example 2: Area of Triangle with Three Sides
6:11
Example 3: Area of Triangle with Three Sides, No Heron's Formula
8:50
Extra Example 1: Area of Triangle with Two Sides and One Angle
-1
Extra Example 2: Area of Triangle with Two Sides and One Angle
-2
Word Problems and Applications of Trigonometry

34m 25s

Intro
0:00
Formulas to Remember
0:11
SOHCAHTOA
0:15
Law of Sines
0:55
Law of Cosines
1:48
Heron's Formula
2:46
Example 1: Telephone Pole Height
4:01
Example 2: Bridge Length
7:48
Example 3: Area of Triangular Field
14:20
Extra Example 1: Kite Height
-1
Extra Example 2: Roads to a Town
-2
Section 9: Systems of Equations and Inequalities
Systems of Linear Equations

55m 40s

Intro
0:00
Introduction
0:04
Graphs as Location of 'True'
1:49
All Locations that Make the Function True
2:25
Understand the Relationship Between Solutions and the Graph
3:43
Systems as Graphs
4:07
Equations as Lines
4:20
Intersection Point
5:19
Three Possibilities for Solutions
6:17
Independent
6:24
Inconsistent
6:36
Dependent
7:06
Solving by Substitution
8:37
Solve for One Variable
9:07
Substitute into the Second Equation
9:34
Solve for Both Variables
10:12
What If a System is Inconsistent or Dependent?
11:08
No Solutions
11:25
Infinite Solutions
12:30
Solving by Elimination
13:56
Example
14:22
Determining the Number of Solutions
16:30
Why Elimination Makes Sense
17:25
Solving by Graphing Calculator
19:59
Systems with More than Two Variables
23:22
Example 1
25:49
Example 2
30:22
Example 3
34:11
Example 4
38:55
Example 5
46:01
(Non-) Example 6
53:37
Systems of Linear Inequalities

1h 13s

Intro
0:00
Introduction
0:04
Inequality Refresher-Solutions
0:46
Equation Solutions vs. Inequality Solutions
1:02
Essentially a Wide Variety of Answers
1:35
Refresher--Negative Multiplication Flips
1:43
Refresher--Negative Flips: Why?
3:19
Multiplication by a Negative
3:43
The Relationship Flips
3:55
Refresher--Stick to Basic Operations
4:34
Linear Equations in Two Variables
6:50
Graphing Linear Inequalities
8:28
Why It Includes a Whole Section
8:43
How to Show The Difference Between Strict and Not Strict Inequalities
10:08
Dashed Line--Not Solutions
11:10
Solid Line--Are Solutions
11:24
11:42
Example of Using a Point
12:41
13:14
Graphing a System
14:53
Set of Solutions is the Overlap
15:17
Example
15:22
Solutions are Best Found Through Graphing
18:05
Linear Programming-Idea
19:52
Use a Linear Objective Function
20:15
Variables in Objective Function have Constraints
21:24
Linear Programming-Method
22:09
Rearrange Equations
22:21
Graph
22:49
Critical Solution is at the Vertex of the Overlap
23:40
Try Each Vertice
24:35
Example 1
24:58
Example 2
28:57
Example 3
33:48
Example 4
43:10
Nonlinear Systems

41m 1s

Intro
0:00
Introduction
0:06
Substitution
1:12
Example
1:22
Elimination
3:46
Example
3:56
Elimination is Less Useful for Nonlinear Systems
4:56
Graphing
5:56
Using a Graphing Calculator
6:44
Number of Solutions
8:44
Systems of Nonlinear Inequalities
10:02
Graph Each Inequality
10:06
Dashed and/or Solid
10:18
11:14
Example 1
13:24
Example 2
15:50
Example 3
22:02
Example 4
29:06
Example 4, cont.
33:40
Section 10: Vectors and Matrices
Vectors

1h 9m 31s

Intro
0:00
Introduction
0:10
Magnitude of the Force
0:22
Direction of the Force
0:48
Vector
0:52
Idea of a Vector
1:30
How Vectors are Denoted
2:00
Component Form
3:20
Angle Brackets and Parentheses
3:50
Magnitude/Length
4:26
Denoting the Magnitude of a Vector
5:16
Direction/Angle
7:52
Always Draw a Picture
8:50
Component Form from Magnitude & Angle
10:10
Scaling by Scalars
14:06
Unit Vectors
16:26
Combining Vectors - Algebraically
18:10
Combining Vectors - Geometrically
19:54
Resultant Vector
20:46
Alternate Component Form: i, j
21:16
The Zero Vector
23:18
Properties of Vectors
24:20
No Multiplication (Between Vectors)
28:30
Dot Product
29:40
Motion in a Medium
30:10
Fish in an Aquarium Example
31:38
More Than Two Dimensions
33:12
More Than Two Dimensions - Magnitude
34:18
Example 1
35:26
Example 2
38:10
Example 3
45:48
Example 4
50:40
Example 4, cont.
56:07
Example 5
1:01:32
Dot Product & Cross Product

35m 20s

Intro
0:00
Introduction
0:08
Dot Product - Definition
0:42
Dot Product Results in a Scalar, Not a Vector
2:10
Example in Two Dimensions
2:34
Angle and the Dot Product
2:58
The Dot Product of Two Vectors is Deeply Related to the Angle Between the Two Vectors
2:59
Proof of Dot Product Formula
4:14
Won't Directly Help Us Better Understand Vectors
4:18
Dot Product - Geometric Interpretation
4:58
We Can Interpret the Dot Product as a Measure of How Long and How Parallel Two Vectors Are
7:26
Dot Product - Perpendicular Vectors
8:24
If the Dot Product of Two Vectors is 0, We Know They are Perpendicular to Each Other
8:54
Cross Product - Definition
11:08
Cross Product Only Works in Three Dimensions
11:09
Cross Product - A Mnemonic
12:16
The Determinant of a 3 x 3 Matrix and Standard Unit Vectors
12:17
Cross Product - Geometric Interpretations
14:30
The Right-Hand Rule
15:17
Cross Product - Geometric Interpretations Cont.
17:00
Example 1
18:40
Example 2
22:50
Example 3
24:04
Example 4
26:20
Bonus Round
29:18
Proof: Dot Product Formula
29:24
Proof: Dot Product Formula, cont.
30:38
Matrices

54m 7s

Intro
0:00
Introduction
0:08
Definition of a Matrix
3:02
Size or Dimension
3:58
Square Matrix
4:42
Denoted by Capital Letters
4:56
When are Two Matrices Equal?
5:04
Examples of Matrices
6:44
Rows x Columns
6:46
7:48
We Use Capitals to Denote a Matrix and Lower Case to Denotes Its Entries
8:32
Using Entries to Talk About Matrices
10:08
Scalar Multiplication
11:26
Scalar = Real Number
11:34
Example
12:36
13:08
Example
14:22
Matrix Multiplication
15:00
Example
18:52
Matrix Multiplication, cont.
19:58
Matrix Multiplication and Order (Size)
25:26
Make Sure Their Orders are Compatible
25:27
Matrix Multiplication is NOT Commutative
28:20
Example
30:08
Special Matrices - Zero Matrix (0)
32:48
Zero Matrix Has 0 for All of its Entries
32:49
Special Matrices - Identity Matrix (I)
34:14
Identity Matrix is a Square Matrix That Has 1 for All Its Entries on the Main Diagonal and 0 for All Other Entries
34:15
Example 1
36:16
Example 2
40:00
Example 3
44:54
Example 4
50:08
Determinants & Inverses of Matrices

47m 12s

Intro
0:00
Introduction
0:06
Not All Matrices Are Invertible
1:30
What Must a Matrix Have to Be Invertible?
2:08
Determinant
2:32
The Determinant is a Real Number Associated With a Square Matrix
2:38
If the Determinant of a Matrix is Nonzero, the Matrix is Invertible
3:40
Determinant of a 2 x 2 Matrix
4:34
Think in Terms of Diagonals
5:12
Minors and Cofactors - Minors
6:24
Example
6:46
Minors and Cofactors - Cofactors
8:00
Cofactor is Closely Based on the Minor
8:01
Alternating Sign Pattern
9:04
Determinant of Larger Matrices
10:56
Example
13:00
Alternative Method for 3x3 Matrices
16:46
Not Recommended
16:48
Inverse of a 2 x 2 Matrix
19:02
Inverse of Larger Matrices
20:00
Using Inverse Matrices
21:06
When Multiplied Together, They Create the Identity Matrix
21:24
Example 1
23:45
Example 2
27:21
Example 3
32:49
Example 4
36:27
Finding the Inverse of Larger Matrices
41:59
General Inverse Method - Step 1
43:25
General Inverse Method - Step 2
43:27
General Inverse Method - Step 2, cont.
43:27
General Inverse Method - Step 3
45:15
Using Matrices to Solve Systems of Linear Equations

58m 34s

Intro
0:00
Introduction
0:12
Augmented Matrix
1:44
We Can Represent the Entire Linear System With an Augmented Matrix
1:50
Row Operations
3:22
Interchange the Locations of Two Rows
3:50
Multiply (or Divide) a Row by a Nonzero Number
3:58
Add (or Subtract) a Multiple of One Row to Another
4:12
Row Operations - Keep Notes!
5:50
Suggested Symbols
7:08
Gauss-Jordan Elimination - Idea
8:04
Gauss-Jordan Elimination - Idea, cont.
9:16
Reduced Row-Echelon Form
9:18
Gauss-Jordan Elimination - Method
11:36
Begin by Writing the System As An Augmented Matrix
11:38
Gauss-Jordan Elimination - Method, cont.
13:48
Cramer's Rule - 2 x 2 Matrices
17:08
Cramer's Rule - n x n Matrices
19:24
Solving with Inverse Matrices
21:10
Solving Inverse Matrices, cont.
25:28
The Mighty (Graphing) Calculator
26:38
Example 1
29:56
Example 2
33:56
Example 3
37:00
Example 3, cont.
45:04
Example 4
51:28
Section 11: Alternate Ways to Graph
Parametric Equations

53m 33s

Intro
0:00
Introduction
0:06
Definition
1:10
Plane Curve
1:24
The Key Idea
2:00
Graphing with Parametric Equations
2:52
Same Graph, Different Equations
5:04
How Is That Possible?
5:36
Same Graph, Different Equations, cont.
5:42
Here's Another to Consider
7:56
Same Plane Curve, But Still Different
8:10
A Metaphor for Parametric Equations
9:36
Think of Parametric Equations As a Way to Describe the Motion of An Object
9:38
Graph Shows Where It Went, But Not Speed
10:32
Eliminating Parameters
12:14
Rectangular Equation
12:16
Caution
13:52
Creating Parametric Equations
14:30
Interesting Graphs
16:38
Graphing Calculators, Yay!
19:18
Example 1
22:36
Example 2
28:26
Example 3
37:36
Example 4
41:00
Projectile Motion
44:26
Example 5
47:00
Polar Coordinates

48m 7s

Intro
0:00
Introduction
0:04
Polar Coordinates Give Us a Way To Describe the Location of a Point
0:26
Polar Equations and Functions
0:50
Plotting Points with Polar Coordinates
1:06
The Distance of the Point from the Origin
1:09
The Angle of the Point
1:33
Give Points as the Ordered Pair (r,θ)
2:03
Visualizing Plotting in Polar Coordinates
2:32
First Way We Can Plot
2:39
Second Way We Can Plot
2:50
First, We'll Look at Visualizing r, Then θ
3:09
Rotate the Length Counter-Clockwise by θ
3:38
Alternatively, We Can Visualize θ, Then r
4:06
'Polar Graph Paper'
6:17
Horizontal and Vertical Tick Marks Are Not Useful for Polar
6:42
Use Concentric Circles to Helps Up See Distance From the Pole
7:08
Can Use Arc Sectors to See Angles
7:57
Multiple Ways to Name a Point
9:17
Examples
9:30
For Any Angle θ, We Can Make an Equivalent Angle
10:44
Negative Values for r
11:58
If r Is Negative, We Go In The Direction Opposite the One That The Angle θ Points Out
12:22
Another Way to Name the Same Point: Add π to θ and Make r Negative
13:44
Converting Between Rectangular and Polar
14:37
Rectangular Way to Name
14:43
Polar Way to Name
14:52
The Rectangular System Must Have a Right Angle Because It's Based on a Rectangle
15:08
Connect Both Systems Through Basic Trigonometry
15:38
Equation to Convert From Polar to Rectangular Coordinate Systems
16:55
Equation to Convert From Rectangular to Polar Coordinate Systems
17:13
Converting to Rectangular is Easy
17:20
Converting to Polar is a Bit Trickier
17:21
Draw Pictures
18:55
Example 1
19:50
Example 2
25:17
Example 3
31:05
Example 4
35:56
Example 5
41:49
Polar Equations & Functions

38m 16s

Intro
0:00
Introduction
0:04
Equations and Functions
1:16
Independent Variable
1:21
Dependent Variable
1:30
Examples
1:46
Always Assume That θ Is In Radians
2:44
Graphing in Polar Coordinates
3:29
Graph is the Same Way We Graph 'Normal' Stuff
3:32
Example
3:52
Graphing in Polar - Example, Cont.
6:45
Tips for Graphing
9:23
Notice Patterns
10:19
Repetition
13:39
Graphing Equations of One Variable
14:39
Converting Coordinate Types
16:16
Use the Same Conversion Formulas From the Previous Lesson
16:23
Interesting Graphs
17:48
Example 1
18:03
Example 2
18:34
Graphing Calculators, Yay!
19:07
Plot Random Things, Alter Equations You Understand, Get a Sense for How Polar Stuff Works
19:11
Check Out the Appendix
19:26
Example 1
21:36
Example 2
28:13
Example 3
34:24
Example 4
35:52
Section 12: Complex Numbers and Polar Coordinates
Polar Form of Complex Numbers

40m 43s

Intro
0:00
Polar Coordinates
0:49
Rectangular Form
0:52
Polar Form
1:25
R and Theta
1:51
Polar Form Conversion
2:27
R and Theta
2:35
Optimal Values
4:05
Euler's Formula
4:25
Multiplying Two Complex Numbers in Polar Form
6:10
Multiply r's Together and Add Exponents
6:32
Example 1: Convert Rectangular to Polar Form
7:17
Example 2: Convert Polar to Rectangular Form
13:49
Example 3: Multiply Two Complex Numbers
17:28
Extra Example 1: Convert Between Rectangular and Polar Forms
-1
Extra Example 2: Simplify Expression to Polar Form
-2
DeMoivre's Theorem

57m 37s

Intro
0:00
Introduction to DeMoivre's Theorem
0:10
n nth Roots
3:06
DeMoivre's Theorem: Finding nth Roots
3:52
Relation to Unit Circle
6:29
One nth Root for Each Value of k
7:11
Example 1: Convert to Polar Form and Use DeMoivre's Theorem
8:24
Example 2: Find Complex Eighth Roots
15:27
Example 3: Find Complex Roots
27:49
Extra Example 1: Convert to Polar Form and Use DeMoivre's Theorem
-1
Extra Example 2: Find Complex Fourth Roots
-2
Section 13: Counting & Probability
Counting

31m 36s

Intro
0:00
Introduction
0:08
Combinatorics
0:56
Definition: Event
1:24
Example
1:50
Visualizing an Event
3:02
Branching line diagram
3:06
3:40
Example
4:18
Multiplication Principle
5:42
Example
6:24
Pigeonhole Principle
8:06
Example
10:26
Draw Pictures
11:06
Example 1
12:02
Example 2
14:16
Example 3
17:34
Example 4
21:26
Example 5
25:14
Permutations & Combinations

44m 3s

Intro
0:00
Introduction
0:08
Permutation
0:42
Combination
1:10
Towards a Permutation Formula
2:38
How Many Ways Can We Arrange the Letters A, B, C, D, and E?
3:02
Towards a Permutation Formula, cont.
3:34
Factorial Notation
6:56
Symbol Is '!'
6:58
Examples
7:32
Permutation of n Objects
8:44
Permutation of r Objects out of n
9:04
What If We Have More Objects Than We Have Slots to Fit Them Into?
9:46
Permutation of r Objects Out of n, cont.
10:28
Distinguishable Permutations
14:46
What If Not All Of the Objects We're Permuting Are Distinguishable From Each Other?
14:48
Distinguishable Permutations, cont.
17:04
Combinations
19:04
Combinations, cont.
20:56
Example 1
23:10
Example 2
26:16
Example 3
28:28
Example 4
31:52
Example 5
33:58
Example 6
36:34
Probability

36m 58s

Intro
0:00
Introduction
0:06
Definition: Sample Space
1:18
Event = Something Happening
1:20
Sample Space
1:36
Probability of an Event
2:12
Let E Be An Event and S Be The Corresponding Sample Space
2:14
'Equally Likely' Is Important
3:52
Fair and Random
5:26
Interpreting Probability
6:34
How Can We Interpret This Value?
7:24
We Can Represent Probability As a Fraction, a Decimal, Or a Percentage
8:04
One of Multiple Events Occurring
9:52
Mutually Exclusive Events
10:38
What If The Events Are Not Mutually Exclusive?
12:20
Taking the Possibility of Overlap Into Account
13:24
An Event Not Occurring
17:14
Complement of E
17:22
Independent Events
19:36
Independent
19:48
Conditional Events
21:28
What Is The Events Are Not Independent Though?
21:30
Conditional Probability
22:16
Conditional Events, cont.
23:51
Example 1
25:27
Example 2
27:09
Example 3
28:57
Example 4
30:51
Example 5
34:15
Section 14: Conic Sections
Parabolas

41m 27s

Intro
0:00
What is a Parabola?
0:20
Definition of a Parabola
0:29
Focus
0:59
Directrix
1:15
Axis of Symmetry
3:08
Vertex
3:33
Minimum or Maximum
3:44
Standard Form
4:59
Horizontal Parabolas
5:08
Vertex Form
5:19
Upward or Downward
5:41
Example: Standard Form
6:06
Graphing Parabolas
8:31
Shifting
8:51
Example: Completing the Square
9:22
Symmetry and Translation
12:18
Example: Graph Parabola
12:40
Latus Rectum
17:13
Length
18:15
Example: Latus Rectum
18:35
Horizontal Parabolas
18:57
Not Functions
20:08
Example: Horizontal Parabola
21:21
Focus and Directrix
24:11
Horizontal
24:48
Example 1: Parabola Standard Form
25:12
Example 2: Graph Parabola
30:00
Example 3: Graph Parabola
33:13
Example 4: Parabola Equation
37:28
Circles

21m 3s

Intro
0:00
What are Circles?
0:08
Example: Equidistant
0:17
0:32
Equation of a Circle
0:44
Example: Standard Form
1:11
Graphing Circles
1:47
Example: Circle
1:56
Center Not at Origin
3:07
Example: Completing the Square
3:51
Example 1: Equation of Circle
6:44
11:51
15:08
Example 4: Equation of Circle
16:57
Ellipses

46m 51s

Intro
0:00
What Are Ellipses?
0:11
Foci
0:23
Properties of Ellipses
1:43
Major Axis, Minor Axis
1:47
Center
1:54
Length of Major Axis and Minor Axis
3:21
Standard Form
5:33
Example: Standard Form of Ellipse
6:09
Vertical Major Axis
9:14
Example: Vertical Major Axis
9:46
Graphing Ellipses
12:51
Complete the Square and Symmetry
13:00
Example: Graphing Ellipse
13:16
Equation with Center at (h, k)
19:57
Horizontal and Vertical
20:14
Difference
20:27
Example: Center at (h, k)
20:55
Example 1: Equation of Ellipse
24:05
Example 2: Equation of Ellipse
27:57
Example 3: Equation of Ellipse
32:32
Example 4: Graph Ellipse
38:27
Hyperbolas

38m 15s

Intro
0:00
What are Hyperbolas?
0:12
Two Branches
0:18
Foci
0:38
Properties
2:00
Transverse Axis and Conjugate Axis
2:06
Vertices
2:46
Length of Transverse Axis
3:14
Distance Between Foci
3:31
Length of Conjugate Axis
3:38
Standard Form
5:45
Vertex Location
6:36
Known Points
6:52
Vertical Transverse Axis
7:26
Vertex Location
7:50
Asymptotes
8:36
Vertex Location
8:56
Rectangle
9:28
Diagonals
10:29
Graphing Hyperbolas
12:58
Example: Hyperbola
13:16
Equation with Center at (h, k)
16:32
Example: Center at (h, k)
17:21
Example 1: Equation of Hyperbola
19:20
Example 2: Equation of Hyperbola
22:48
Example 3: Graph Hyperbola
26:05
Example 4: Equation of Hyperbola
36:29
Conic Sections

18m 43s

Intro
0:00
Conic Sections
0:16
Double Cone Sections
0:24
Standard Form
1:27
General Form
1:37
Identify Conic Sections
2:16
B = 0
2:50
X and Y
3:22
Identify Conic Sections, Cont.
4:46
Parabola
5:17
Circle
5:51
Ellipse
6:31
Hyperbola
7:10
Example 1: Identify Conic Section
8:01
Example 2: Identify Conic Section
11:03
Example 3: Identify Conic Section
11:38
Example 4: Identify Conic Section
14:50
Section 15: Sequences, Series, & Induction
Introduction to Sequences

57m 45s

Intro
0:00
Introduction
0:06
Definition: Sequence
0:28
Infinite Sequence
2:08
Finite Sequence
2:22
Length
2:58
Formula for the nth Term
3:22
Defining a Sequence Recursively
5:54
Initial Term
7:58
Sequences and Patterns
10:40
First, Identify a Pattern
12:52
How to Get From One Term to the Next
17:38
Tips for Finding Patterns
19:52
More Tips for Finding Patterns
24:14
Even More Tips
26:50
Example 1
30:32
Example 2
34:54
Fibonacci Sequence
34:55
Example 3
38:40
Example 4
45:02
Example 5
49:26
Example 6
51:54
Introduction to Series

40m 27s

Intro
0:00
Introduction
0:06
Definition: Series
1:20
Why We Need Notation
2:48
Simga Notation (AKA Summation Notation)
4:44
Thing Being Summed
5:42
Index of Summation
6:21
Lower Limit of Summation
7:09
Upper Limit of Summation
7:23
Sigma Notation, Example
7:36
Sigma Notation for Infinite Series
9:08
How to Reindex
10:58
How to Reindex, Expanding
12:56
How to Reindex, Substitution
16:46
Properties of Sums
19:42
Example 1
23:46
Example 2
25:34
Example 3
27:12
Example 4
29:54
Example 5
32:06
Example 6
37:16
Arithmetic Sequences & Series

31m 36s

Intro
0:00
Introduction
0:05
Definition: Arithmetic Sequence
0:47
Common Difference
1:13
Two Examples
1:19
Form for the nth Term
2:14
Recursive Relation
2:33
Towards an Arithmetic Series Formula
5:12
Creating a General Formula
10:09
General Formula for Arithmetic Series
14:23
Example 1
15:46
Example 2
17:37
Example 3
22:21
Example 4
24:09
Example 5
27:14
Geometric Sequences & Series

39m 27s

Intro
0:00
Introduction
0:06
Definition
0:48
Form for the nth Term
2:42
Formula for Geometric Series
5:16
Infinite Geometric Series
11:48
Diverges
13:04
Converges
14:48
Formula for Infinite Geometric Series
16:32
Example 1
20:32
Example 2
22:02
Example 3
26:00
Example 4
30:48
Example 5
34:28
Mathematical Induction

49m 53s

Intro
0:00
Introduction
0:06
Belief Vs. Proof
1:22
A Metaphor for Induction
6:14
The Principle of Mathematical Induction
11:38
Base Case
13:24
Inductive Step
13:30
Inductive Hypothesis
13:52
A Remark on Statements
14:18
Using Mathematical Induction
16:58
Working Example
19:58
Finding Patterns
28:46
Example 1
30:17
Example 2
37:50
Example 3
42:38
The Binomial Theorem

1h 13m 13s

Intro
0:00
Introduction
0:06
We've Learned That a Binomial Is An Expression That Has Two Terms
0:07
Understanding Binomial Coefficients
1:20
Things We Notice
2:24
What Goes In the Blanks?
5:52
Each Blank is Called a Binomial Coefficient
6:18
The Binomial Theorem
6:38
Example
8:10
The Binomial Theorem, cont.
10:46
We Can Also Write This Expression Compactly Using Sigma Notation
12:06
Proof of the Binomial Theorem
13:22
Proving the Binomial Theorem Is Within Our Reach
13:24
Pascal's Triangle
15:12
Pascal's Triangle, cont.
16:12
16:24
Zeroth Row
18:04
First Row
18:12
Why Do We Care About Pascal's Triangle?
18:50
Pascal's Triangle, Example
19:26
Example 1
21:26
Example 2
24:34
Example 3
28:34
Example 4
32:28
Example 5
37:12
Time for the Fireworks!
43:38
Proof of the Binomial Theorem
43:44
We'll Prove This By Induction
44:04
Proof (By Induction)
46:36
Proof, Base Case
47:00
Proof, Inductive Step - Notation Discussion
49:22
Induction Step
49:24
Proof, Inductive Step - Setting Up
52:26
Induction Hypothesis
52:34
What We What To Show
52:44
Proof, Inductive Step - Start
54:18
Proof, Inductive Step - Middle
55:38
Expand Sigma Notations
55:48
Proof, Inductive Step - Middle, cont.
58:40
Proof, Inductive Step - Checking In
1:01:08
Let's Check In With Our Original Goal
1:01:12
Want to Show
1:01:18
Lemma - A Mini Theorem
1:02:18
Proof, Inductive Step - Lemma
1:02:52
Proof of Lemma: Let's Investigate the Left Side
1:03:08
Proof, Inductive Step - Nearly There
1:07:54
Proof, Inductive Step - End!
1:09:18
Proof, Inductive Step - End!, cont.
1:11:01
Section 16: Preview of Calculus
Idea of a Limit

40m 22s

Intro
0:00
Introduction
0:05
Motivating Example
1:26
Fuzzy Notion of a Limit
3:38
Limit is the Vertical Location a Function is Headed Towards
3:44
Limit is What the Function Output is Going to Be
4:15
Limit Notation
4:33
Exploring Limits - 'Ordinary' Function
5:26
Test Out
5:27
Graphing, We See The Answer Is What We Would Expect
5:44
Exploring Limits - Piecewise Function
6:45
If We Modify the Function a Bit
6:49
Exploring Limits - A Visual Conception
10:08
Definition of a Limit
12:07
If f(x) Becomes Arbitrarily Close to Some Number L as x Approaches Some Number c, Then the Limit of f(x) As a Approaches c is L.
12:09
We Are Not Concerned with f(x) at x=c
12:49
We Are Considering x Approaching From All Directions, Not Just One Side
13:10
Limits Do Not Always Exist
15:47
Finding Limits
19:49
Graphs
19:52
Tables
21:48
Precise Methods
24:53
Example 1
26:06
Example 2
27:39
Example 3
30:51
Example 4
33:11
Example 5
37:07
Formal Definition of a Limit

57m 11s

Intro
0:00
Introduction
0:06
New Greek Letters
2:42
Delta
3:14
Epsilon
3:46
Sometimes Called the Epsilon-Delta Definition of a Limit
3:56
Formal Definition of a Limit
4:22
What does it MEAN!?!?
5:00
The Groundwork
5:38
Set Up the Limit
5:39
The Function is Defined Over Some Portion of the Reals
5:58
The Horizontal Location is the Value the Limit Will Approach
6:28
The Vertical Location L is Where the Limit Goes To
7:00
The Epsilon-Delta Part
7:26
The Hard Part is the Second Part of the Definition
7:30
Second Half of Definition
10:04
Restrictions on the Allowed x Values
10:28
The Epsilon-Delta Part, cont.
13:34
Sherlock Holmes and Dr. Watson
15:08
The Adventure of the Delta-Epsilon Limit
15:16
Setting
15:18
We Begin By Setting Up the Game As Follows
15:52
The Adventure of the Delta-Epsilon, cont.
17:24
17:46
What If I Try Larger?
19:39
Technically, You Haven't Proven the Limit
20:53
Here is the Method
21:18
What We Should Concern Ourselves With
22:20
Investigate the Left Sides of the Expressions
25:24
We Can Create the Following Inequalities
28:08
Finally…
28:50
Nothing Like a Good Proof to Develop the Appetite
30:42
Example 1
31:02
Example 1, cont.
36:26
Example 2
41:46
Example 2, cont.
47:50
Finding Limits

32m 40s

Intro
0:00
Introduction
0:08
Method - 'Normal' Functions
2:04
The Easiest Limits to Find
2:06
It Does Not 'Break'
2:18
It Is Not Piecewise
2:26
Method - 'Normal' Functions, Example
3:38
Method - 'Normal' Functions, cont.
4:54
The Functions We're Used to Working With Go Where We Expect Them To Go
5:22
5:42
Method - Canceling Factors
7:18
One Weird Thing That Often Happens is Dividing By 0
7:26
Method - Canceling Factors, cont.
8:16
Notice That The Two Functions Are Identical With the Exception of x=0
8:20
Method - Canceling Factors, cont.
10:00
Example
10:52
Method - Rationalization
12:04
Rationalizing a Portion of Some Fraction
12:05
Conjugate
12:26
Method - Rationalization, cont.
13:14
Example
13:50
Method - Piecewise
16:28
The Limits of Piecewise Functions
16:30
Example 1
17:42
Example 2
18:44
Example 3
20:20
Example 4
22:24
Example 5
24:24
Example 6
27:12
Continuity & One-Sided Limits

32m 43s

Intro
0:00
Introduction
0:06
Motivating Example
0:56
Continuity - Idea
2:14
Continuous Function
2:18
All Parts of Function Are Connected
2:28
Function's Graph Can Be Drawn Without Lifting Pencil
2:36
There Are No Breaks or Holes in Graph
2:56
Continuity - Idea, cont.
3:38
We Can Interpret the Break in the Continuity of f(x) as an Issue With the Function 'Jumping'
3:52
Continuity - Definition
5:16
A Break in Continuity is Caused By the Limit Not Matching Up With What the Function Does
5:18
Discontinuous
6:02
Discontinuity
6:10
Continuity and 'Normal' Functions
6:48
Return of the Motivating Example
8:14
One-Sided Limit
8:48
One-Sided Limit - Definition
9:16
Only Considers One Side
9:20
Be Careful to Keep Track of Which Symbol Goes With Which Side
10:06
One-Sided Limit - Example
10:50
There Does Not Necessarily Need to Be a Connection Between Left or Right Side Limits
11:16
Normal Limits and One-Sided Limits
12:08
Limits of Piecewise Functions
14:12
'Breakover' Points
14:22
We Find the Limit of a Piecewise Function By Checking If the Left and Right Side Limits Agree With Each Other
15:34
Example 1
16:40
Example 2
18:54
Example 3
22:00
Example 4
26:36
Limits at Infinity & Limits of Sequences

32m 49s

Intro
0:00
Introduction
0:06
Definition: Limit of a Function at Infinity
1:44
A Limit at Infinity Works Very Similarly to How a Normal Limit Works
2:38
Evaluating Limits at Infinity
4:08
Rational Functions
4:17
Examples
4:30
For a Rational Function, the Question Boils Down to Comparing the Long Term Growth Rates of the Numerator and Denominator
5:22
There are Three Possibilities
6:36
Evaluating Limits at Infinity, cont.
8:08
Does the Function Grow Without Bound? Will It 'Settle Down' Over Time?
10:06
10:26
Limit of a Sequence
12:20
What Value Does the Sequence Tend to Do in the Long-Run?
12:41
The Limit of a Sequence is Very Similar to the Limit of a Function at Infinity
12:52
Numerical Evaluation
14:16
Numerically: Plug in Numbers and See What Comes Out
14:24
Example 1
16:42
Example 2
21:00
Example 3
22:08
Example 4
26:14
Example 5
28:10
Example 6
31:06
Instantaneous Slope & Tangents (Derivatives)

51m 13s

Intro
0:00
Introduction
0:08
The Derivative of a Function Gives Us a Way to Talk About 'How Fast' the Function If Changing
0:16
Instantaneous Slop
0:22
Instantaneous Rate of Change
0:28
Slope
1:24
The Vertical Change Divided by the Horizontal
1:40
Idea of Instantaneous Slope
2:10
What If We Wanted to Apply the Idea of Slope to a Non-Line?
2:14
Tangent to a Circle
3:52
What is the Tangent Line for a Circle?
4:42
Tangent to a Curve
5:20
Towards a Derivative - Average Slope
6:36
Towards a Derivative - Average Slope, cont.
8:20
An Approximation
11:24
Towards a Derivative - General Form
13:18
Towards a Derivative - General Form, cont.
16:46
An h Grows Smaller, Our Slope Approximation Becomes Better
18:44
Towards a Derivative - Limits!
20:04
Towards a Derivative - Limits!, cont.
22:08
We Want to Show the Slope at x=1
22:34
Towards a Derivative - Checking Our Slope
23:12
Definition of the Derivative
23:54
Derivative: A Way to Find the Instantaneous Slope of a Function at Any Point
23:58
Differentiation
24:54
Notation for the Derivative
25:58
The Derivative is a Very Important Idea In Calculus
26:04
The Important Idea
27:34
Why Did We Learn the Formal Definition to Find a Derivative?
28:18
Example 1
30:50
Example 2
36:06
Example 3
40:24
The Power Rule
44:16
Makes It Easier to Find the Derivative of a Function
44:24
Examples
45:04
n Is Any Constant Number
45:46
Example 4
46:26
Area Under a Curve (Integrals)

45m 26s

Intro
0:00
Introduction
0:06
Integral
0:12
Idea of Area Under a Curve
1:18
Approximation by Rectangles
2:12
The Easiest Way to Find Area is With a Rectangle
2:18
Various Methods for Choosing Rectangles
4:30
Rectangle Method - Left-Most Point
5:12
The Left-Most Point
5:16
Rectangle Method - Right-Most Point
5:58
The Right-Most Point
6:00
Rectangle Method - Mid-Point
6:42
Horizontal Mid-Point
6:48
Rectangle Method - Maximum (Upper Sum)
7:34
Maximum Height
7:40
Rectangle Method - Minimum
8:54
Minimum Height
9:02
Evaluating the Area Approximation
10:08
Split the Interval Into n Sub-Intervals
10:30
More Rectangles, Better Approximation
12:14
The More We Us , the Better Our Approximation Becomes
12:16
Our Approximation Becomes More Accurate as the Number of Rectangles n Goes Off to Infinity
12:44
Finding Area with a Limit
13:08
If This Limit Exists, It Is Called the Integral From a to b
14:08
The Process of Finding Integrals is Called Integration
14:22
The Big Reveal
14:40
The Integral is Based on the Antiderivative
14:46
The Big Reveal - Wait, Why?
16:28
The Rate of Change for the Area is Based on the Height of the Function
16:50
Height is the Derivative of Area, So Area is Based on the Antiderivative of Height
17:50
Example 1
19:06
Example 2
22:48
Example 3
29:06
Example 3, cont.
35:14
Example 4
40:14
Section 17: Appendix: Graphing Calculators

10m 41s

Intro
0:00
0:06
Should I Get a Graphing Utility?
0:20
Free Graphing Utilities - Web Based
0:38
Personal Favorite: Desmos
0:58
Free Graphing Utilities - Offline Programs
1:18
GeoGebra
1:31
Microsoft Mathematics
1:50
Grapher
2:18
Other Graphing Utilities - Tablet/Phone
2:48
Should You Buy a Graphing Calculator?
3:22
The Only Real Downside
4:10
4:20
If You Plan on Continuing in Math and/or Science
4:26
If Money is Not Particularly Tight for You
4:32
If You Don't Plan to Continue in Math and Science
5:02
If You Do Plan to Continue and Money Is Tight
5:28
5:44
Which Graphing Calculator is Best?
5:46
Too Many Factors
5:54
6:12
The Old Standby
7:10
TI-83 (Plus)
7:16
TI-84 (Plus)
7:18
9:17
9:19
9:35
10:09
Graphing Calculator Basics

10m 51s

Intro
0:00
0:06
Skim It
0:20
Play Around and Experiment
0:34
Syntax
0:40
Definition of Syntax in English and Math
0:46
Pay Careful Attention to Your Syntax When Working With a Calculator
2:08
Make Sure You Use Parentheses to Indicate the Proper Order of Operations
2:16
3:54
Settings
4:58
You'll Almost Never Need to Change the Settings on Your Calculator
5:00
Tell Calculator In Settings Whether the Angles Are In Radians or Degrees
5:26
Graphing Mode
6:32
Error Messages
7:10
Don't Panic
7:11
Internet Search
7:32
So Many Things
8:14
More Powerful Than You Realize
8:18
Other Things Your Graphing Calculator Can Do
8:24
Playing Around
9:16
Graphing Functions, Window Settings, & Table of Values

10m 38s

Intro
0:00
Graphing Functions
0:18
Graphing Calculator Expects the Variable to Be x
0:28
Syntax
0:58
The Syntax We Choose Will Affect How the Function Graphs
1:00
Use Parentheses
1:26
The Viewing Window
2:00
One of the Most Important Ideas When Graphing Is To Think About The Viewing Window
2:01
For Example
2:30
The Viewing Window, cont.
2:36
Window Settings
3:24
Manually Choose Window Settings
4:20
x Min
4:40
x Max
4:42
y Min
4:44
y Max
4:46
Changing the x Scale or y Scale
5:08
Window Settings, cont.
5:44
Table of Values
7:38
Allows You to Quickly Churn Out Values for Various Inputs
7:42
For example
7:44
Changing the Independent Variable From 'Automatic' to 'Ask'
8:50
Finding Points of Interest

9m 45s

Intro
0:00
Points of Interest
0:06
Interesting Points on the Graph
0:11
Roots/Zeros (Zero)
0:18
Relative Minimums (Min)
0:26
Relative Maximums (Max)
0:32
Intersections (Intersection)
0:38
Finding Points of Interest - Process
1:48
Graph the Function
1:49
2:12
Choose Point of Interest Type
2:54
Identify Where Search Should Occur
3:04
Give a Guess
3:36
Get Result
4:06
5:10
Find Out What Input Value Causes a Certain Output
5:12
For Example
5:24
7:18
Derivative
7:22
Integral
7:30
But How Do You Show Work?
8:20
Parametric & Polar Graphs

7m 8s

Intro
0:00
Change Graph Type
0:08
Located in General 'Settings'
0:16
Graphing in Parametric
1:06
Set Up Both Horizontal Function and Vertical Function
1:08
For Example
2:04
Graphing in Polar
4:00
For Example
4:28
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• ## Related Books

 1 answerLast reply by: Professor Selhorst-JonesMon Mar 24, 2014 10:09 AMPost by Sarang Janakiraman on March 23, 2014On # 3 of Example 1, can we also say the limit is -âˆž? That's how I learnt it but would you find that acceptable? 1 answerLast reply by: Professor Selhorst-JonesMon Nov 4, 2013 3:26 PMPost by edick safarians on November 4, 2013If you were to graph these functions, what feature would appear in the graph at the limiting value? Why? 1 answerLast reply by: Professor Selhorst-JonesSun Jul 28, 2013 8:28 PMPost by Jason Todd on July 27, 2013Do you have any videos for Calculus, Differential equations, and/or Linear algebra somewhere else? I'm reviewing all of my math for engineering and your videos are a God-send.

### Limits at Infinity & Limits of Sequences

• Important Idea: infinity is not a location. It is the idea of going on forever, moving on to ever larger numbers. You can travel towards ∞, but you can never reach ∞.
• We denote the limit of a function at infinity with
 lim f(x)               and lim f(x).
They mean the value that f(x) approaches as x goes off toward −∞ or ∞, respectively.
• A limit at infinity works very similarly to how a normal limit works. Does the function "settle down" to a specific value L? It's just in this case, we're asking about the long-term behavior instead of x→ c. [It is important to note that most of the functions we're used to working with do not have limits at infinity.]
• The type of functions we will work with most often that can have a limit at infinity are rational functions. We worked with these many lessons ago when we learned about asymptotes. They are functions created from a fraction with polynomials in the numerator and denominator:
 f(x) = an xn + an−1 xn−1 + …+ a1 x + a0bm xm + bm−1 xm−1 + …+ b1 x + b0 .
Notice that n is the numerator's degree and m is the denominator's degree. Then we can find limits at infinity by the below:
• If n < m     ⇒     limx→ ±∞ f(x)  = 0
• If n=m     ⇒     limx→ ±∞ f(x)  = [(an)/(bm)]
• If n > m     ⇒     limx→ ±∞ f(x)  does not exist
• In general, when trying to find limits at infinity, think in terms of how the function will be affected as x grows very large (positive and negative). Does the function grow without bound? Will it "settle down" over time? Two good ways to think about this are:
• What happens if we plug in a LARGE NUMBER?
• What are the rates of growth in the function? Which parts grow faster? Which parts grow slower?
By thinking through these questions, you can get a good idea of how the function will behave over the long-term.
• We can take this idea of a limit going to infinity and apply it to a sequence as well. Let a1, a2, a3,...   be some infinite sequence. Then we can consider
 lim an.
The limit of a sequence is very similar to the limit of a function at infinity. Does the sequence "settle down" to a specific value L as n→ ∞?
• Sometimes it can be difficult to tell how a function or sequence will behave in the long-run. In that case, we can evaluate the function numerically: plug in numbers and see what comes out. By using a calculator, we can plug in very large numbers (positive and negative) and see what happens to the function or sequence. Doing this will give us a good sense of the long-term behavior. Similarly, if you have access to a graphing calculator or program, you can graph the function. Expand the viewing window to a large horizontal region and look to see if the graph "settles down" in the long-run.

### Limits at Infinity & Limits of Sequences

Complete the table below to approximate the limit at infinity.

lim
x→ ∞
27x

3x2−5x

 x
 10
 100
 1000
 10 000
 100 000
 f(x)
• The limit at infinity is a way of talking about what value the function approaches as x heads off towards infinity-that is, as x gets very, very large. [Notice that not all functions will have a limit at infinity.]
• One way to find the value of this limit (or if it even exists) is by using larger and larger values for x. If plugging in really big values for x makes the function "settle down" towards a specific value, then we have found the value of the limit. If, on the other hand, larger and larger values give significantly different numbers each time, then the limit does not exist.
• For this problem, we're finding the value of the limit by trying out some very large numbers. We plug in various large numbers, using an even bigger one each time. To complete the table, just plug each number in to the function, then use a calculator to get an approximate value. For example, the first two x-values would work as follows:
 x=10     ⇒ 27(10) 3(10)2 −5(10) ≈     1.080  000

 x=100     ⇒ 27(100) 3(100)2 −5(100) ≈     0.091 525
• Continuing this process with each of the x-values, the completed table is
 x
 10
 100
 1000
 10 000
 100 000
 f(x)
 1.080  000
 0.091 525
 0.009 015
 0.000  900
 0.000 090
Thus, looking at the table, we see that as x→ ∞, the value of the function is shrinking to extremely tiny numbers. As x→ ∞, we see f(x) → 0. Therefore our infinite limit is   limx→ ∞  [27x/(3x2−5x)]    =   0.
 x
 10
 100
 1000
 10 000
 100 000
 f(x)
 1.080  000
 0.091 525
 0.009 015
 0.000  900
 0.000 090
limx→ ∞  [27x/(3x2−5x)]    =   0
Use a graphing calculator/program to graph the function and approximate the limit at infinity.
 limx → − ∞ 6x2+2x 3x2−7x+5
• The limit at infinity is a way of talking about what value the function approaches as x heads off towards infinity-that is, as x gets very, very large. [Notice that not all functions will have a limit at infinity.] For this problem, we're considering the limit as x → −∞, so we're considering what happens to the function as x goes to very, very large negative numbers.
• One way to find the value of this limit (or if it even exists) is by using a graphing calculator to graph the function. Using a graph of the function, we can see if the function "settles down" as x→ −∞. Looking at the graph very far to the left, if the function seems to be heading towards a specific value the limit will be that value. On the other hand, if the graph never "settles down" and just keeps growing or changing, then the limit does not exist.
• Do what the problem tells you to do and graph the function with a graphing calculator/program. You'll get something like the below.
• Notice that, while interesting, it's not quite as useful as we'd like. We're interested in x→ − ∞, but the left-most value is a paltry x=−15. If we're going to be confident, we should look much farther to the left. Instead, change the Window Settings so that the minimum x-value is much more distant. Something like x=−100 or x=−1000. While we're at it, notice that we don't really care what happens on the right side of the graph, so we can use a much smaller value over there. Finally, don't set your y-values to be too large: from our initial graph, we can see that the limit (if it exists) probably won't get up too high. With this in mind, set the y-minimum at −1 and the y-maximum at 5. Having changed the Window Settings, graph the function again. Now you can use Trace to find values or draw a horizontal line to see what number the function is approaching as it goes far to the left.
limx → − ∞  [(6x2+2x)/(3x2−7x+5)]    =   2
Evaluate the limit (if it exists):    limx → ∞  [3/(x−7)]
• The limit at infinity is a way of talking about what value the function approaches as x heads off towards infinity-that is, as x gets very, very large. This means that we want to think in these terms: imagine x becoming very large and think about what effect it has on the rest of the function.
• From the lesson, we learned that if the function we are trying to evaluate is set up in the form of a rational function (polynomial numerator and denominator), we can figure out the infinite limit based on the degrees of the top and bottom. Both the numerator of 3 and the denominator of x−7 are polynomials, so we can do this here. Notice that 3 has a degree of 0 (since it has no variable), while x−7 has a degree of 1 (since it has the variable raised to the first power: x=x1). Thus we have that the numerator's degree is less than the denominator's degree (0 < 1), so the lesson told us that the infinite limit comes out to 0.
• An alternative way to do this is to think in terms of "What happens to the function as we use BIG NUMBERS for x?" We can symbolize this idea of BIG NUMBERS by using ∞. [Notice that ∞ is not actually a number. It is the idea of having BIG NUMBERS that grow forever, not a value we can actually plug in. Because of this, the below is not formal mathematics. We are playing fast and loose with the math here: while it will make sense to us, this method may not be acceptable in every single situation. Most teachers will accept it, but it is not guaranteed.] With this in mind, we plug in our BIG NUMBER (as ∞) and see what happens:
 limx → ∞ 3 x−7 ⇒ 3 ∞−7
Here, notice two things: 1) ∞ minus any constant is unaffected since any number is tiny compared to ∞, so we can throw out the constant; 2) Dividing any constant by ∞ turns to 0: any number is tiny compared to ∞, so it will crush any constant in division.
 3 ∞−7 ⇒ 3 ∞ ⇒     0
limx → ∞  [3/(x−7)]   =  0
Evaluate the limit (if it exists):    limx → ∞  [(1−5x)/(x+4)]
• The limit at infinity is a way of talking about what value the function approaches as x heads off towards infinity-that is, as x gets very, very large. This means that we want to think in these terms: imagine x becoming very large and think about what effect it has on the rest of the function.
• From the lesson, we learned that if the function we are trying to evaluate is set up in the form of a rational function (polynomial numerator and denominator), we can figure out the infinite limit based on the degrees of the top and bottom. Both the numerator of 1−5x and the denominator of x+4 are polynomials, so we can do this here. Notice that 1−5x has a degree of 1 (x=x1), as does x+4. Thus we have that the numerator's degree equals the denominator's degree (1=1). At this point, the lesson said that we need to make a ratio out of the coefficients attached to the variable with the highest exponent:
 1−5x   ⇒  Coefficient: −5                      x+4   ⇒  Coefficient: 1
Now we make a ratio of dividing the numerator's coefficient by the denominator's coefficient to find the limit:
 −5 1 =     −5
• An alternative way to do this is to think in terms of "What happens to the function as we use BIG NUMBERS for x?" We can symbolize this idea of BIG NUMBERS by using ∞. [Notice that ∞ is not actually a number. It is the idea of having BIG NUMBERS that grow forever, not a value we can actually plug in. Because of this, the below is not formal mathematics. We are playing fast and loose with the math here: while it will make sense to us, this method may not be acceptable in every single situation. Most teachers will accept it, but it is not guaranteed.] With this in mind, we plug in our BIG NUMBER (as ∞) and see what happens:
 limx → ∞ 1−5x x+4 ⇒ 1−5·∞∞+4
Now notice that ∞ plus or minus any constant is unaffected since any number is tiny compared to ∞, so we can throw out the constant.
 1−5·∞∞+4 ⇒ −5 ·∞∞
Finally, remember that the ∞ on the top and the ∞ on the bottom represent the same BIG NUMBER so we can cancel them out:
 −5 ·∞∞ ⇒     −5
limx → ∞  [(1−5x)/(x+4)]    =   −5
Evaluate the limit (if it exists):    limx → −∞  [(x2)/(100x−70)]
• The limit at infinity is a way of talking about what value the function approaches as x heads off towards infinity-that is, as x gets very, very large. For this problem, we're working with x→ − ∞ so we want to think in these terms: imagine x becoming a very large, negative number and think about what effect it has on the rest of the function.
• From the lesson, we learned that if the function we are trying to evaluate is set up in the form of a rational function (polynomial numerator and denominator), we can figure out the infinite limit based on the degrees of the top and bottom. The numerator of x2 and the denominator of 100x−70 are polynomials, so we can do this here. Notice that x2 has a degree of 2 (x2), while 100x−70 has a degree of 1 (x=x1). Thus we have that the numerator's degree is greater than the denominator's degree (2 > 1), so the lesson told us that the limit at infinity does not exist.
• An alternative way to do this is to think in terms of "What happens to the function as we use BIG NUMBERS for x?" We can symbolize this idea of BIG NUMBERS by using ∞. [Notice that ∞ is not actually a number. It is the idea of having BIG NUMBERS that grow forever, not a value we can actually plug in. Because of this, the below is not formal mathematics. We are playing fast and loose with the math here: while it will make sense to us, this method may not be acceptable in every single situation. Most teachers will accept it, but it is not guaranteed.] With this in mind, we plug in our BIG NUMBER (as ∞) and see what happens:
 limx → −∞ x2100x−70 ⇒ (−∞)2100·(−∞)−70 ⇒ ∞2−100·∞− 70
Now notice that ∞ plus or minus any constant is unaffected since any number is tiny compared to ∞, so we can throw out the constant.
 ∞2−100·∞− 70 ⇒ ∞2−100 ·∞
Furthermore, notice that we can split ∞2 into ∞·∞, then cancel out one of them:
 ∞2−100 ·∞ ⇒ ∞·∞−100 ·∞ ⇒ ∞−100
Finally, dividing ∞ by any constant will have no effect: ∞ is just too big to notice being divided by a normal number. (Still, the negative will have an effect.)
 ∞−100 ⇒     − ∞100 ⇒     − ∞
At this point we're done, but we need to interpret what −∞ means. Remember that ∞ is a stand-in for ever growing BIG NUMBERS. Thus, because the result will always have growing BIG NUMBERS, it is never "settling down" to a single value. Instead, as the function goes farther and farther left, it continues to go down forever. This means that the limit does not exist.
limx → −∞  [(x2)/(100x−70)] does not exist
Evaluate the limit (if it exists):    limx → ∞  [(4x3−7)/(x2−2x3)]
• The limit at infinity is a way of talking about what value the function approaches as x heads off towards infinity-that is, as x gets very, very large. This means that we want to think in these terms: imagine x becoming very large and think about what effect it has on the rest of the function.
• From the lesson, we learned that if the function we are trying to evaluate is set up in the form of a rational function (polynomial numerator and denominator), we can figure out the infinite limit based on the degrees of the top and bottom. Both the numerator of 4x3−7 and the denominator of x2−2x3 are polynomials, so we can do this here. Notice that 4x3−7 has a degree of 3 (x3), as does x2−2x3. Thus we have that the numerator's degree equals the denominator's degree (3=3). At this point, the lesson said that we need to make a ratio out of the coefficients attached to the variable with the highest exponent:
 4x3−7   ⇒  Coefficient: 4                      x2−2x3   ⇒  Coefficient: −2
Now we make a ratio of dividing the numerator's coefficient by the denominator's coefficient to find the limit:
 4 −2 =     −2
• An alternative way to do this is to think in terms of "What happens to the function as we use BIG NUMBERS for x?" We can symbolize this idea of BIG NUMBERS by using ∞. [Notice that ∞ is not actually a number. It is the idea of having BIG NUMBERS that grow forever, not a value we can actually plug in. Because of this, the below is not formal mathematics. We are playing fast and loose with the math here: while it will make sense to us, this method may not be acceptable in every single situation. Most teachers will accept it, but it is not guaranteed.] With this in mind, we plug in our BIG NUMBER (as ∞) and see what happens:
 limx → ∞ 4x3−7 x2−2x3 ⇒ 4(∞)3−7 (∞)2−2(∞)3
Now notice that ∞ plus or minus any constant is unaffected since any number is tiny compared to ∞, so we can throw out the constant. Along similar lines of thought, but considerably more advanced, notice that ∞2 is massively smaller than ∞3. In a way, because we have a larger "class" of ∞, we can throw out the smaller scale one that is being added/subtracted.
 4·∞3−7 ∞2−2·∞3 ⇒ 4·∞3−2·∞3
Finally, remember that the ∞ on the top and the ∞ on the bottom represent the same BIG NUMBER so we can cancel them out:
 4·∞3−2·∞3 ⇒ 4 −2 ⇒     −2
limx → ∞  [(4x3−7)/(x2−2x3)]    =   −2
Evaluate the limit (if it exists):    limx → −∞  cos( [1/x])
• The limit at infinity is a way of talking about what value the function approaches as x heads off towards infinity-that is, as x gets very, very large. For this problem, we're working with x→ − ∞ so we want to think in these terms: imagine x becoming a very large, negative number and think about what effect it has on the rest of the function.
• In the lesson, we did not specifically learn how to deal with trigonometric functions like cosine. However, we did learn that
 limx → −∞ 1 x =     0
With this idea in mind, we see that since we have cosine acting on [1/x], when we consider the limit at infinity for cos([1/x]), we can swap out the [1/x] for what it becomes in the long run:
 limx → −∞ cos ⎛⎝ 1 x ⎞⎠ = limx → −∞ cos( 0)
From there, we see that cos(0) is unaffected by the limit (because it no longer contains the variable x), and we can just evaluate it as normal.
 limx → −∞ cos( 0)     =     cos( 0)     =     1
• An alternative way to do this is to think in terms of "What happens to the function as we use BIG NUMBERS for x?" We can symbolize this idea of BIG NUMBERS by using ∞. [Notice that ∞ is not actually a number. It is the idea of having BIG NUMBERS that grow forever, not a value we can actually plug in. Because of this, the below is not formal mathematics. We are playing fast and loose with the math here: while it will make sense to us, this method may not be acceptable in every single situation. Most teachers will accept it, but it is not guaranteed.] With this in mind, we plug in our BIG NUMBER (as ∞) and see what happens:
 limx → −∞ cos ⎛⎝ 1 x ⎞⎠ ⇒     cos ⎛⎝ 1 −∞ ⎞⎠ ⇒     cos(0)     ⇒     1
limx → −∞  cos( [1/x])    =   1
Evaluate the limit (if it exists):    limx → ∞  [ [3x/(2x−1)] + [(10x2+3)/((2x−3)2)]]
• The limit at infinity is a way of talking about what value the function approaches as x heads off towards infinity-that is, as x gets very, very large. This means that we want to think in these terms: imagine x becoming very large and think about what effect it has on the rest of the function.
• Before we try to evaluate the limit, it will make it a little bit easier to expand things so we can better apply the ideas we've learned:
 limx → ∞ ⎡⎣ 3x 2x−1 + 10x2+3 (2x−3)2 ⎤⎦ = limx → ∞ ⎡⎣ 3x 2x−1 + 10x2+3 4x2−12x+9 ⎤⎦
• Now that we clearly have two rational function expressions (polynomial numerator and denominator), we can apply what we learned before. Deal with each part of the sum separately:

lim
x → ∞
3x

2x−1
⇒
 Num. degree:
 1
 Denom. degree:
 1
⇒
 N. coefficient:
 3
 D. coefficient:
 2
Since the numerator and denominator have matching polynomial degrees, we find the value of the limit by taking the ratio of the leading coefficients:
 limx → ∞ 3x 2x−1 = 3 2
Next, the other half of the limit:

lim
x → ∞
10x2+3

4x2−12x+9
⇒
 Num. degree:
 2
 Denom. degree:
 2
⇒
 N. coefficient:
 10
 D. coefficient:
 4
Since the numerator and denominator have matching polynomial degrees, we find the value of the limit by taking the ratio of the leading coefficients:
 limx → ∞ 10x2+3 4x2−12x+9 = 10 4 = 5 2
Finally, since we know the infinite limit of each half on its own, we can just plug those in when we take the limit of them together:
 limx → ∞ ⎡⎣ 3x 2x−1 + 10x2+3 4x2−12x+9 ⎤⎦ = 3 2 + 5 2 = 8 2 =     4
• An alternative way to do this is to think in terms of "What happens to the function as we use BIG NUMBERS for x?" We can symbolize this idea of BIG NUMBERS by using ∞. [Notice that ∞ is not actually a number. It is the idea of having BIG NUMBERS that grow forever, not a value we can actually plug in. Because of this, the below is not formal mathematics. We are playing fast and loose with the math here: while it will make sense to us, this method may not be acceptable in every single situation. Most teachers will accept it, but it is not guaranteed.] With this in mind, we plug in our BIG NUMBER (as ∞) and see what happens:
 limx → ∞ ⎡⎣ 3x 2x−1 + 10x2+3 4x2−12x+9 ⎤⎦ ⇒ 3·∞2·∞−1 + 10 ·∞2 + 3 4 ·∞2 −12 ·∞+ 9
For each fraction, notice that ∞ plus or minus any constant is unaffected since any number is tiny compared to ∞, so we can throw out the constant. Along similar lines of thought, but considerably more advanced, notice that ∞ is massively smaller than ∞2. In a way, because we have a larger "scale" of ∞, we can throw out the smaller scale one that is being added/subtracted in the second fraction:
 3·∞2·∞−1 + 10 ·∞2 + 3 4 ·∞2 −12 ·∞+ 9 ⇒ 3 ·∞2 ·∞ + 10 ·∞24 ·∞2
Now we can cancel out common factors of ∞ in each fraction since they just represent the same BIG NUMBER, then simplify:
 3 ·∞2 ·∞ + 10 ·∞24 ·∞2 ⇒ 3 2 + 10 4 = 3 2 + 5 2 = 8 2 =     4
limx → ∞  [ [3x/(2x−1)] + [(10x2+3)/((2x−3)2)]]   =   4
Consider the sequence with general term an = [((n+2)!)/(n! ·2n)]. What is the limit of the sequence as n→ ∞?
• Although this question is based around a sequence, it works almost identically to previous problems about limits at infinity. We want to think about what happens to the sequence terms as n heads off towards infinity-that is, as n gets very, very large. Imagine n becoming very large and think about what effect it will have.
• In the current form, it's a little hard to tell what's going to happen because we have factorials on the top and bottom. Begin by simplifying the general term by canceling out based on how factorials work (if you don't remember factorials, check out the lesson Permutations and Combinations):
 an = (n+2)! n! ·2n = (n+2)(n+1)n! n! ·2n = (n+2)(n+1) 2n = n2 +3n +22n
• In this form, we can more easily think about how the sequence behaves as n→ ∞. Two approaches we could use are plugging in large numbers or graphing the expression: either one of these methods will help us figure out what happens in the long run as n gets very, very large. In previous problems, we've relied on working with rational functions (polynomial numerator and denominator) to figure out infinite limits, but we can't do that here. Since the denominator is 2n, we don't have a polynomial on the bottom, so we can't use the exact same analysis.
• However, we can still think in terms of "What happens to the expression as we use BIG NUMBERS for n?" We can symbolize this idea of BIG NUMBERS by using ∞. [Notice that ∞ is not actually a number. It is the idea of having BIG NUMBERS that grow forever, not a value we can actually plug in. Because of this, the below is not formal mathematics. We are playing fast and loose with the math here: while it will make sense to us, this method may not be acceptable in every single situation. Most teachers will accept it, but it is not guaranteed.] With this in mind, we plug in our BIG NUMBER (as ∞) and see what happens:
 limn → ∞ n2 +3n +2 2n ⇒ ∞2 + 3·∞+ 2 2∞
Begin by noticing that, for the numerator, the biggest "number" by far is ∞2. Because it is ∞·∞, it is much bigger than 3·∞ and 2, so we can just throw those away without affecting anything:
 ∞2 + 3·∞+ 2 2∞ ⇒ ∞22∞
This now brings us to the challenging question: which is bigger, ∞2 or 2? Well, remember, ∞ isn't actually a number: it's the idea of plugging in BIG NUMBERS. With this in mind, we realize that we really want to know which grows faster, n2 or 2n? Here we might think back to the lesson on Exponential Functions and remember that 2n grows extremely quickly (remember the story with the grains of rice on the chessboard from that lesson). While n2 gets big, it is completely outclassed by 2n, and will be "crushed" into nothing. Even if we don't remember our work with exponential functions, we can still think about what happens as the numbers get big. Consider if we used a fairly small "big" number, like n=100: while 1002 is pretty large, it is absolutely tiny in comparison to 2100 (check it with a calculator if you're not sure). This disparity in size will only continue to widen as n gets larger, so we see that the denominator will "crush" the numerator into nothing in the long run. Therefore, we have
 ∞22∞ ⇒     0
limn → ∞  [((n+2)!)/(n! ·2n)]   =   0
A publishing house is getting ready to print a book. In order to do so, they must first prepare the book (edit, design, etc.), which will cost a total of $12 000. Once the book is prepared, it will cost the publishing house$1.70 per book they print. (a) What is the average cost per book if they print 10  000 books? What if they print 100  000 books? (b) Give a formula that determines the average cost per book of printing n books. (c) Determine the limit of the average cost per book as n → ∞. Interpret what this means in terms of the problem.
• In general, the average is determined as
 Average     = Total amount of "stuff" Number of "objects"
For this problem, the "stuff" is the cost of printing all the books, while the number of "objects" is how many books are printed.
 Average cost per book     = Total cost of making booksNumber of books made
• (a): Calculate the cost of producing 10 000 books. Remember, there is an initial, flat cost of $12 000, then an additional$1.70 per book. This means:
 Cost of producing 10 000 books     =     12 000 + (10 000)(1.70)     =     29 000
Once we know the cost of producing 10 000 books, we want to know the average cost per book, so we divide the total cost by the number of books:
 Average cost per book, printing 10 000     = 29 000 10 000 =     2.90
Thus when we print 10 000 books, the average cost is $2.90 per book. We follow the same method to do it for 100 000 books:  Cost of producing 100 000 books = 12 000 + (100 000)(1.70) = 182 000  Average cost per book, printing 100 000 = 182 000 100 000 = 1.82 Thus when we print 100 000 books, the average cost is$1.82 per book.
• (b): To find a general formula that determines the cost per book of printing n books, notice that we are determining our average cost per book from the idea
 Average cost per book     = Total cost of making booksNumber of books made
Thus, to figure out the formula, we just need to calculate the 1) Cost of making n books; and 2) Number of books made. Well, as we saw in the above step, there is an initial, flat cost of $12 000, then an additional$1.70 per book. Since we are making n books, we have
 Cost of making n books     =     12 000 + 1.70·n
It's much easier to figure out the number of books made: we're making n books, so that's the number made. Put these two things together to find the average cost per book:
 Average cost per book, printing n     = 12 000 +1.70 ·nn
• (c): Since we now have a formula for the average cost per book, we just need to consider the limit as n→ ∞:
 limn → ∞ 12 000 +1.70 ·n n
From the previous problems we did in this lesson, we saw that this will just come down to a ratio of the coefficients on the variables:
 limn → ∞ 12 000 +1.70 ·n n = 1.70 1 =     1.70
Thus, as n→ ∞, the average cost per book goes to $1.70. Interpreting this, remember what this all means. The limit as n→ ∞ is a way to ask how much the average cost per book is as we produce HUGE numbers of books. Thinking about it like that, we realize that as the number of books printed becomes extremely large, the initial start-up cost becomes a tiny fraction of the overall cost. Once we're printing on the scale of billions of books, that initial cost to prepare the book becomes negligible in comparison. With huge numbers of books, we no longer need to pay attention to the start-up cost, so it's just a matter of how much it costs to print additional books. Since printing another book costs$1.70, that is what the average cost per book becomes when printing in massive quantities.
(a) $2.90,$1.82 (b) [(12 000 +1.70 ·n)/n] (c) \$1.70-When printing very large numbers of books, the start-up cost becomes negligible, so the only important thing is the cost of producing each book.

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

### Limits at Infinity & Limits of Sequences

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

• Intro 0:00
• Introduction 0:06
• Definition: Limit of a Function at Infinity 1:44
• A Limit at Infinity Works Very Similarly to How a Normal Limit Works
• Evaluating Limits at Infinity 4:08
• Rational Functions
• Examples
• For a Rational Function, the Question Boils Down to Comparing the Long Term Growth Rates of the Numerator and Denominator
• There are Three Possibilities
• Evaluating Limits at Infinity, cont.
• Does the Function Grow Without Bound? Will It 'Settle Down' Over Time?
• Limit of a Sequence 12:20
• What Value Does the Sequence Tend to Do in the Long-Run?
• The Limit of a Sequence is Very Similar to the Limit of a Function at Infinity
• Numerical Evaluation 14:16
• Numerically: Plug in Numbers and See What Comes Out
• Example 1 16:42
• Example 2 21:00
• Example 3 22:08
• Example 4 26:14
• Example 5 28:10
• Example 6 31:06

### Transcription: Limits at Infinity & Limits of Sequences

Hi--welcome back to Educator.com.0000

Today, we are going to talk about limits at infinity and limits of sequences.0002

Up until this point, we have only considered the idea of a limit as x goes to c, where c is some fixed horizontal location.0006

But what if, instead of focusing on a single location, we considered what would happen to the function if x just kept traveling forever off to infinity?0013

What would we do if our x wasn't going to a single place, but we were just sort of watching it ride off into the sunset?0020

This is the idea that we will consider in this lesson.0026

In fact, we have actually considered this many lessons ago, in the lesson on horizontal asymptotes,0029

because a horizontal asymptote was the question of what this function goes to in the very, very long run;0034

as x runs off to infinity (both the positive and the negative infinity)0040

what vertical value, what y-value, what function value, do we end up running to in the long run?0046

That was the idea of a horizontal asymptote; we will draw on those ideas in this lesson, as well.0052

Before we start, though, let's remind ourselves of something important: infinity is not a location.0056

It is not some place; you don't run to the end of the rainbow, and there you are at infinity; you always have to keep running.0061

The idea of infinity is just going on forever, moving on to ever-larger numbers.0068

You never actually reach infinity; you can travel towards infinity, but you can never reach infinity.0074

It is just the idea of going on forever; it is the idea of riding into the sunset.0081

You don't ever actually make it to the sun; you just keep riding off into the distance.0085

That is the idea of what it means to travel towards infinity.0090

You never actually arrive there; x can't be equal to infinity; but we can consider the idea of what happens as x runs off forever and ever and ever.0093

That is what we will be thinking about.0101

We denote the limit of a function at infinity (even though it says "at," we are really meaning "as it goes towards")--0104

the limit as x goes to negative infinity of f(x) and the limit as x goes to positive infinity of f(x)...0111

And this means the value that f(x) approaches as x goes off to either negative infinity or positive infinity, respectively.0118

So, negative infinity, with an actual negative sign--that means negative infinity.0127

And infinity, just on its own with nothing there--we just assume that there is a positive sign in front of it, even though we don't see it.0133

If you don't see a symbol, it is assumed that we are talking about positive infinity, as opposed to talking about negative infinity.0140

Negative infinity would go off to the left, whereas positive infinity, or just infinity with no symbol in it, goes off to the right, forever.0146

OK, a limit at infinity works very similarly to how a normal limit works.0154

Does the function settle down--does it go to some specific value l?0160

It is just that, in this case, we are talking about long-term behavior, instead of x going to some specific horizontal location.0166

Instead of what happens to the function as x gets close to c, it is what happens to the function as it rides off into the sunset.0173

What happens as it goes off to some infinity?0179

Now, it is important to note that most of the functions--in fact, the vast majority of the functions0181

that we are used to working with--do not have limits at infinity.0186

For example, if we consider f(x) = x, one of the simplest functions that we are used to using,0189

that one just keeps growing forever; so it has no limit at infinity.0194

It doesn't stabilize; it doesn't settle down to some value.0199

If you plug in 1 million, you get 1 million out of it; if you plug in 1 billion, you get 1 billion out of it; if you plug in 1 trillion, you get 1 trillion out of it.0204

As you keep plugging in larger and larger numbers, it is just going to keep growing and growing and growing and growing.0211

It is never going to stop growing, which means it is never going to settle down, which means it is never going to some specific value l.0215

Most of the functions that we are used to dealing with on a daily basis are actually not going to have limits at infinity,0222

because they never settle down to a specific value--they grow without bound.0227

They might grow off to positive infinity; they might grow off to negative infinity (that is, vertically--what the output goes off to be).0231

But they are growing without bound; they aren't going to some specific value, so that means that they won't have limits at infinity.0237

Still, there are definitely functions that do have limits at infinity.0243

The type of functions that we will work with most often (there are some others that won't be this, but)0247

the ones that we will work with most often, that have limits at infinity, are rational functions.0252

We worked with these many lessons ago, when we learned about asymptotes.0256

They are functions of the form f(x) = n(x)/d(x), where n(x) and d(x) are polynomials, and d(x) is not equal to 0.0260

So, you probably remember these things like this: g(x) = (3x - 1)/(x3 + 4), h(x) = 1/x4,0271

j(x) = (x5 + 47x2)/(x3 - 15)--just some polynomial divided by some other polynomial.0280

Now, because we are dividing by something, that means that our denominator, what we are dividing by,0289

has the possibility to grow faster than the numerator.0294

Basically, our denominator can grow fast enough to keep the numerator in check--to keep that numerator from blowing off and just going on forever.0297

The denominator can actually grow faster and keep it down.0305

It has the ability to stabilize it to a single value in the long run.0309

And that is why we end up seeing rational functions give us limits at infinity so often.0312

For a rational function, the question is basically comparing the long-term growth rates of the numerator and the denominator.0317

It is a question of which is growing faster over the long term: is it the numerator or the denominator?0323

If the numerator is growing faster than the denominator over the long term, then the thing is not really going to settle down,0329

because it is just going to keep getting bigger and bigger and bigger.0335

If, on the other hand, the denominator is growing faster than the numerator,0338

then the denominator will crush the numerator, and so it will be forced to settle down.0341

Now, we already studied this idea in horizontal asymptotes; and so, let's look at those results.0346

If we have some rational function, f(x) (and notice that that is just some polynomial divided by some polynomial--0352

some constant times xn, some other constant times xn - 1, and working our way0358

down to a constant times x, plus some constant, and the same thing on the bottom, as well--0364

it is just constant times x to some value, constant times x to that value minus 1,0369

and working our way down to a constant; so it is some polynomial over some polynomial),0374

notice, from this, that n is the numerator's degree; we have xn as the largest exponent on the top;0378

and m is the denominator's degree, so m is the biggest exponent on the bottom.0387

From this, there are three possibilities: if n is less than m, then that means that our top,0394

the exponent in our numerator, n, is going to be less than the exponent in our denominator,0401

which means that the denominator is going to grow faster, so it will crush the numerator, causing us to have a horizontal asymptote of y = 0.0408

If, on the other hand, n equals m (the leading exponent on the numerator is equal0415

to the leading exponent on the denominator), they grow in the same category of speed.0423

They won't necessarily have precisely the same; but one of them won't massively outclass the other one.0427

At that point, what we will do is compare the leading coefficients, an and bm.0432

The horizontal asymptote that we get out of that is a ratio of the leading coefficients, an divided by bm,0440

because in the long run, since we have the same exponent on top and bottom,0447

that part, the x to the some exponent, will grow at the same rate on the top and the bottom.0451

So, it will end up just being a question of what number they are multiplying in front.0455

And that is why we get a horizontal asymptote based on that.0458

And finally, the last one: if n is greater than n (that is, the leading exponent on the numerator is greater0461

than the leading exponent on our denominator), then that means that the numerator will be able to run faster0467

than the denominator and escape the denominator's ability to bound it and hold it back.0471

And so, it will just go off forever, and it won't be able to stabilize to a single value, which means that it will have no horizontal asymptote.0476

We can write this in a way where we can talk about this as limits at infinity, because,0484

since horizontal asymptotes tell us the behavior of f(x) as x goes to positive or negative infinity--0489

a horizontal asymptote is what value it approaches over the long term--they are also telling us the limits at infinity,0496

since the limit at infinity is what value it approaches over the long term.0501

So, for some rational function f(x), let n be the numerator's degree, and m be the denominator's degree.0505

Let an and bm be the leading coefficients of the numerator and the denominator, respectively.0514

Then, we have: if n is less than m, the limit as x goes to positive or negative infinity of f(x) equals 0.0524

The numerator is smaller, effectively; the exponent is smaller, so its growth rate is smaller than the denominator.0532

So, the denominator crushes it down to 0.0538

If n equals m, then the limit as x goes to positive or negative infinity of f(x) is equal to an divided by bm.0541

The growth rate on the top and the bottom is the same, because they have the same leading exponent.0547

So now, it is a question of what the ratio is of the coefficients in front of them.0551

And finally, if n is greater than m, that means that the leading coefficient on top is greater than the leading coefficient on the bottom,0555

which means that the growth rate of the top is greater than the growth rate of the bottom.0560

So, there the top manages to escape and run off forever.0564

So, that means that the limit as x goes to positive or negative infinity of f(x) simply does not exist, because it will never stabilize to a single value.0568

That tells us what to do with rational functions; the previous method allows us to easily find limits at infinity for rational functions.0577

But we will occasionally have to deal with other types of functions, as well.0583

We won't only have to deal with rational functions.0586

So, in that case, the best thing to do--there is no simple formulaic method for how to figure out "Here is what the limit is going to end up being."0589

In this case, what you want to do is think in terms of how the function will be affected as x grows very large, both positively and negatively.0596

You want to think, "Does the function grow without bound?"0605

If it just grows forever and ever, or goes off down forever and ever,0608

then that means that it is going to end up not stabilizing to something in the long term, which means that it won't have a limit at infinity.0612

Or, on the other hand, will it settle down--does it go to some specific value?0617

Does it settle down; does it approach some specific thing over the long term?0621

So, there are two good ways to think about this, to figure out if this is the case.0625

We want to think about what happens if we plug in a large number.0628

And by "large number," I mean to think like...if I were to plug in something on the scale of a million,0632

a billion, a trillion, a really, really, really big number--something large--what would happen to this?0637

You don't have to come up with actual answers to what will happen to the thing.0643

You don't have to produce some number in the end.0647

You just want to think, "If there was a really, really big number here, how would it affect the other things that it is near?"0649

What would no longer really be important? What would still matter?0654

If you were to plug in a really big number, what is going to keep changing?0658

And what will happen as that large number continues to increase?0660

That is one way of looking at it; the other way to look at it is to think, "What are the rates of growth in the function?"0664

Which part of the function is growing faster and will continue to grow?0670

Which parts grow faster, and which parts are growing slower, that get slower and slower as we go farther and farther out?0674

Thinking about these two things (that one will especially help if you are dealing with a fraction--0681

what is the comparison between the growth rate of our numerator versus the growth rate of our denominator?)--0686

thinking in terms of rates of growth, what is growing faster and what is growing slower,0690

how the rate of growth will be affected as we go out to larger and larger x--0694

these sorts of things (that and what would happen if I plugged in a very large number)--0698

thinking in terms of those two ideas will give you a good, intuitive sense of what is going to happen.0702

If you think through these questions, you can get a good idea of where the function will be going,0707

of how the function will behave over the long term.0711

You won't necessarily be able to come up with an absolutely precise answer.0715

But you will be able to get a sense of if it makes sense for this thing to have a limit at all.0717

But sometimes, you will even be able to get an exact answer by thinking through this; it depends on the situation.0721

But just sort of try to be creative and think in a very broad, general sense.0726

We will also talk about ways where, if you are not quite sure how to think about it,0730

there are numerical ways that you can figure out...get a good sense of what is going on.0733

And we will talk about that in just a few slides.0737

Another thing that we can talk about, using this idea of a limit at infinity, as a limit goes to infinity: we can apply it to a sequence, as well.0740

If we have some sequence, a1, a2, a3, a4, a5...0747

so it is some infinite sequence that just keeps going on forever and ever and ever--0752

then what we can consider is the limit as n goes to infinity of an.0756

What does our sequence go towards--what value does the sequence approach in the long run?0760

How does this thing work out? What will it be going towards as n becomes ever larger and larger and larger?0767

The limit of a sequence (that is this thing right here) is very similar to the limit of a function at infinity.0772

The question is, "Does the sequence settle down--does it go to some specific value l as n runs off to infinity?"0779

As our n becomes larger and larger and larger, does our sequence stabilize0787

into something that is going to basically be the same as we go farther and farther and farther?0790

Now, it is important to note that, just like functions, most sequences will not have limits as n goes to infinity.0795

For example, a really simple sequence: 1, 2, 3, 4, 5, 6... has no limit, because it just grows forever.0801

It will just grow forever, because we have 1, 2, 3, 4...so if we look at a very far-out term, it will be very large.0810

But if we look at an even farther-out term, it will have continued to grow, and it will be even larger.0817

So, it is not going to head towards a steady value; it is not going to stabilize and go to some specific value l; it will never settle down.0821

Nonetheless, there are definitely still some sequences out there that will end up stabilizing; and we will see those in the examples.0830

But just because we are looking at the limit as n goes to infinity of a sequence doesn't necessarily mean that it will stabilize.0836

There are plenty of sequences out there that won't stabilize at all.0842

For example, every arithmetic sequence we have ever looked at won't stabilize,0845

because it just continues stepping up and stepping up and stepping up and stepping up.0849

Finally, we can also talk about numerical evaluation.0855

Sometimes, it can be difficult to tell how a function or sequence will behave in the long run.0858

In that case, we can evaluate the function numerically--that is to say, use numbers.0862

We will just plug in numbers, and we will see what comes out.0866

If we have a calculator, we can use a calculator and just plug in very large numbers.0869

And we will want to plug in both positive and negative numbers.0874

And then, see what happens: we just see what happens to our function or a sequence.0876

We plug in 10; then we plug in 100; then we plug in 1000; then we plug in 10000.0880

Does it seem like it is going to a number, or does it seem like it is just growing larger and larger and larger?0884

Doing this will give us a good sense for long-term behavior.0889

We will be able to tell that, yes, it seems to be just growing forever and ever,0893

or it seems to be stabilizing as we go to larger and larger numbers that we are plugging in.0896

So, this gives us a way to numerically get a sense of what is going on.0900

It is not foolproof, but for the most part you will be able to figure out which one it is going to end up going to.0903

And you probably also have a very good estimate of what value it is going to be approaching in the long term.0908

Similarly, if you have access to a graphing calculator or some graphing program, you can graph the function.0913

If you expand the viewing window to a large horizontal region (say -100 to +100),0920

then you can look and see if the graph settles down in the long run.0925

Does it seem like it is being pulled to a single value, or does it seem like it is just blowing forever and going to keep growing forever and ever and ever?0929

Now, once again, it is not a foolproof method; there are some times where the function will fool you for the first 1000x.0935

From 0 to 1000, it will look like it is growing forever and ever.0941

But then, after 1000, it will actually end up steadying out to a single value.0944

But for the most part, this is a pretty good way to see if this is going to end up approaching a single value,0948

or if it is going to just keep growing forever and ever.0953

So, just take a look at the graph; make sure you use a large horizontal region.0955

If you only look at -10 to +10 for x, you might not have a very good sense of what happens in the long run.0960

You want to use a very large horizontal region, like -100 to +100.0965

It might be kind of hard for your graphing calculator to graph as you get to larger and larger windows;0969

but the larger you can deal with, the better, really, because that will tell you a better idea of what is going on.0973

For the most part, though, -100 to 100 should probably do for anything that you want to graph.0978

And then, just look: does the graph settle down--it is going to tend to a single value in the long run?0982

As you go to those larger x-values, is it basically always graphing at the same height?0987

And if that is the case, it probably has a limit; and you can figure out, looking at the graph, about what value that ends up being.0991

All right, we are ready for some examples.0998

The first example: Evaluate the limits below if they exist.1000

The first one is the limit, as x goes to negative infinity, of 1/x.1003

In this case, we want to think about what ends up happening.1007

We have that specific formulaic method; we have a step-by-step thing for analyzing what is going to come out here.1010

We already can see that the answer has to be 0, from that formulaic method.1015

But let's also think about what happens: as x goes off to negative infinity, our 1/x...1018

well, x continues to grow; how does the numerator grow?1025

It doesn't grow; it is just 1--it stays constant; it just sits there as 1 forever and ever and ever.1029

But our x continues to get larger and larger and larger.1033

We plug in 1000 (well, we plug in -1000, because we are going to negative infinity), and we have 1/-1000;1036

we plug in negative 10 thousand, and we have 1 over negative 10 thousand;1043

we plug in negative 1 billion, and we have 1 over negative 1 billion.1045

We are getting smaller and smaller and smaller; it is crushing down to 0, so we see that it ends up being 0.1050

We can also tell that that is going to end up being the case, just because our numerator has a leading exponent of 0.1056

It is x0, which would only give us 1; and our bottom has x1, so the bottom has a higher leading exponent;1062

so that means that it is going to be crushed down to 0 in the long run.1068

Similarly, the limit as x goes to positive infinity of 1/x...well, our bottom is x; it does grow;1072

but our top is 1--it just stays the same forever and ever.1079

So, that means, as x gets larger and larger and larger, that it is going to crush our entire fraction down to 0.1081

So, we end up getting 0 for the limit here.1087

Over here, we have the limit as x goes to negative infinity of (x3 + 3)/(x2 + x).1091

So, notice: in this case, we have 3 as the leading exponent up top.1097

That means that we have this growth rate somewhere in the neighborhood of x3.1102

But on the bottom, we have a leading exponent of squared; so we have a growth rate somewhere in the neighborhood of x2.1106

What that means is that the top, in the long run, will end up growing much, much, much faster than our bottom will.1112

It is going to end up outrunning the bottom, effectively.1119

We can also imagine this: if we were to plug in a very large number,1122

then we would have big cubed, plus 3, over big squared, plus big.1125

Well, notice: the number here that is the most important is big cubed.1135

Big squared is a very large number, but big cubed is going to be even larger.1141

There is a huge difference between 102 and 103; 10 squared is 100, but 10 cubed is 1000.1147

So, as we get out to very large numbers, big cubed is going to be massively larger than big squared.1152

And similarly, big is just to the 1; so it is practically not going to be anything compared to big squared.1157

So, in the long run, the plus 3 doesn't really matter; the big to the 1 doesn't really matter.1162

The big squared doesn't really matter, because the biggest thing of all, by far, is big cubed.1167

So, that means that we are going to get really, really large numbers up top,1172

and nothing else is really going to be of a comparative size.1175

So, that means, in the long run, that it is going to blow out to infinity.1178

In this case, it will blow out to negative infinity; so that means that it is not stabilizing to a single value.1181

So, we say that the limit does not exist, because it is never going to stabilize,1186

because our top grows faster than our bottom will.1190

In this case, it is going to negative infinity, so we might think of it as growing down.1195

But in either case, it is going larger than the bottom will.1199

The final one is the limit as x goes to positive infinity of (8x4 + 3x2)/(2x4 - 17).1202

So, in this case, we see that the important thing is a leading coefficient of 4 on top, and a leading coefficient of 4 on the bottom.1210

3x2 and -17...as we get to very large numbers, as we go farther and farther out towards infinity,1215

3x2 and -17...they aren't really going to matter much in the long run, as we get to very large numbers.1222

So, it really is determined by 8x4/2x4.1227

In that case, we see that the x4 and the x4 are going to effectively cancel each other out.1231

So, all that we really have left, in the long run, is 8/2; 8/2 simplifies to 4, and there is our answer.1235

We can also see this as the leading coefficients, 8 and 2; since we have the leading exponents,1242

we just do the leading coefficient on the top, divided by the leading coefficient on the bottom, 8/2; and that is equal to 4; great.1248

The next example: Let's look at the limits here--these are limits of sequences, since it is n going to infinity.1256

Evaluate the limits below, if they exist: limit as n goes to infinity of 1/n2.1262

Once again, we see that this is n2; up at the top, it is just 1; it is constant.1267

So, our bottom continues to grow and grow and grow and grow, but our top stays the same.1272

So, as we divide by larger and larger and larger numbers, it crushes it down to 0, just like the reasoning that we used previously.1276

The limit as n goes to infinity of (5n - 1)/(n + 4): well, in this case, we have 5n and n over here.1282

The -1 and the 4 don't change as n goes larger and larger and larger.1291

In the long run, we have big numbers for n; -1 and 4 are going to basically have no effect on what is going on.1296

So, we can think of them as not really mattering, which leaves us with 5n/n in the long run;1302

so we are just comparing--what are the two leading coefficients?1307

5/1 = 5; and there is the limit as n goes out to infinity, the limit of the sequence--what happens to the sequence in the long run.1311

Compare the limits below: which limit exists? Why?1323

All right, our first one is the limit as x goes to infinity of sin(x), and our second one is the limit as x goes to infinity of sin(x)/x.1326

OK, let's get a sense for what happens to the limit as x goes to infinity of sin(x).1334

Well, first let's take a quick graph of how sin(x) behaves.1338

We start here at x = 0; it goes up and down and up and down and up and down and up...1342

and it just continues in this method forever and ever and ever and ever.1354

It never changes this thing of going up/down, up/down, up/down; that is how sin(x) works--it repeats itself over and over forever.1358

What that means is that we have it bouncing; we are bouncing between +1 at its maximum and -1, forever.1366

We are always going up/down, up/down, up/down, up/down; we never stop bouncing up and down.1377

So, if that is the case, since we never stop bouncing up and down, it never settles down to a specific value.1384

All right, it is going to always be near the values of +1 and -1 and 0; but it never steadies out to a single thing.1391

If we say that it is going to be at 0 in the long run, well, it is going to end up getting away from 0 over and over and over.1399

So, it is never settling down; if it never settles down, that means that the limit does not exist; the limit here does not exist.1405

What about our other limit, though--the limit as x goes to infinity of sin(x)/x?1417

Well, what happens? Once again, sin(x), our top, bounces between -1 and +1 forever.1423

OK, but the bottom grows forever; this x right here is going to get larger and larger and larger as x goes off to infinity.1437

So, as x goes off to infinity, the bottom will grow forever.1450

The top oscillates between +1 and -1, +1 and -1, +1 and -1; but our bottom gets larger and larger and larger: 1, 10, 100, 1000...1455

So, since the top never really manages to get very far--it isn't growing without bound--1465

it is just bouncing between these two numbers--even at its largest possible values of +1 and -1,1470

if we divide that by x out at a billion, x out at a quadrillion...it is going to be crushing it down to these very small numbers.1475

Thus, we have that the fraction will end up being crushed.1483

The bottom, in the long run, is going to crush the top.1487

In the long run, it ends up looking like 0.1497

If you want to see what that ends up looking like, what we have is this divide by x...well, 1/x has a graph like this, as it approaches it.1500

So, our sin(x) is going to be bouncing between these two possible extremes.1515

So, it ends up getting squeezed down, closer and closer and closer and closer to this 0 value.1520

And that is why it ends up happening--that is why you end up having this long-term value of 0.1526

In sin(x), it bounces up and down forever; it just keeps going up and down and up and down.1532

But over here in sin(x) divided by x, this "divide by x," over the long run, pinches it down--keeps it crushed down.1537

It starts with these large oscillations; but as it goes farther and farther out, it has to get smaller and smaller,1544

because the x, the "divide by x," crushes it down; and so it gets crushed down to a single value.1548

It will continue to oscillate, but it is getting closer and closer and closer; it has to stay in this window near this height value of 0.1553

And so, since it gets crushed down slowly over time to 0, it effectively just approaches 0 in the long run.1560

So, we have a limit as x goes to infinity of 0.1566

The fourth example: Compare the limits below. Which exists? Why?1569

First, we could just graph this to get a sense of what is going on.1574

If we graph this, the limit as x goes to negative infinity of 2x, and the limit as x goes to positive infinity of 2x...1577

well, if we graph, what does 2x look like?1584

Well, at 0, it is going to be at 1; and then as we go out, it is going to get very large very quickly.1586

As we go to the left, it is going to get smaller and smaller and smaller and smaller.1590

All right, that is what happens: it will never get past the x-axis, but it is going to end up getting smaller and smaller and smaller.1594

If we look at some values, we see that at x = -1, x = -2, x = -3...for this, we would have 2-1,1600

and then 2-2, and then 2-3, which would come out to be 1/2, 1/22, so 1/4,1609

1/23, which would be 1/8; so 1/2, 1/4, 1/8...it gets smaller and smaller.1616

It is always going to smaller values as x goes off to negative infinity.1624

Since it is always going off to smaller and smaller values--that is, values closer and closer to 0--1628

as 2 to the negative number becomes very large--it is going to be1633

1 over 2 to the very large number, which is going to make a very tiny fraction.1638

So, over the long run, it ends up getting crushed down to 0.1642

However, if we look at the limit as x goes to positive infinity of 2x,1646

if we look at just the first couple of numbers, 21, and then 22,1650

and then 23 (that is, x = 1, x = 2, x = 3), we would end up getting 2, and then 4, and then 8;1655

so it is getting bigger and bigger as it ends up going larger and larger.1661

As it gets closer and closer to positive infinity, it will get larger and larger and larger.1667

We end up seeing that, since it is going to get larger and larger and larger, it is never going to stabilize to a single value.1671

It is never going to go to some specific value l, so that means that the limit does not exist,1677

because it will just blow off forever and ever, going up forever and ever.1682

The fifth example: Evaluate the limit as x goes to infinity of 2x/(x + 1) - x2/4(x + 1)2.1686

The first thing to notice here is that this portion of this fraction here doesn't really have an effect on this fraction here during the process of the limit.1696

So, as x goes to infinity, this fraction and this fraction don't really interact with each other.1705

They are basically separate; so if they are basically separate, we can split the limit into the two portions.1711

So, we can split it into the limit as x goes to infinity of the first portion, 2x/(x + 1),1716

minus the limit as x goes to infinity of the second portion, x2/4(x + 1)2.1725

All right, now we can evaluate both of these on their own.1736

For the first one, 2x/x...they both have the same leading coefficient.1738

If we imagine very large numbers going in there, we are comparing two times big number, over big number plus 1.1743

The plus one doesn't really matter; so we only care about two times big over big.1748

The "big"s cancel each other out effectively, and we can think of this as just going...it will go to precisely 2 in the long run.1752

As x goes off forever and ever, it is going to get closer and closer to 2.1760

Minus the limit as x goes to infinity...for this one, we are not quite sure, because...1765

let's expand the (x + 1)2, although we can see x21768

divided by something that is also going to contain an x2.1772

So, we should probably be able to see that in the long run, as x goes to positive infinity, we will end up seeing it go to 1/4.1775

But let's expand it, so we can see it clearly: the limit as x goes to infinity...1786

x2 doesn't change on top...divided by 4, times (x + 1)2...that is just equal to x2 + 2x + 1.1789

So, if we multiply 4 times that expansion of (x + 1)2, we get 4x2 + 8x + 4.1800

So, we still have 2 in front, minus the limit, as x goes to infinity, of this.1809

Well, actually, at this point, we don't even need to do another limit,1814

because we can see that the top has a leading exponent of x2; the bottom has a leading exponent of squared, as well.1817

So, we just compare the coefficients in front, 1 and 4.1824

Since we have big number squared up top, divided by 4 times big number squared, plus 8 times big number...1828

that is not really going to be much, compared to big squared...plus just plain 4 (that is not going to be much compared to big),1834

it is really 1 big squared, over 4 big squared; the big squareds effectively cancel out, leaving us with 1/4 in the long run.1839

We have minus 1/4; we have broken down each piece of the limit; we have figured out1847

that the first portion becomes 2; the second portion becomes 1/4; so 2 - 1/4 simplifies to 7/4.1852

All right, the final example, Example 6: Evaluate the limit as n goes to infinity of the sequence (n - 1)!/(n + 1)!.1862

The first thing we want to do here is think, "Well, we don't really see how to do this immediately;1871

so we want to see if we can simplify this into something where we have less going on."1875

Factorials--it is kind of hard to see what is going on with factorials.1879

So, maybe let's get a sense for if there is some way to cancel them and expand things.1882

We realize that they are both based around a somewhat similar thing.1886

n + 1 isn't very far from n - 1; so we can expand the factorials, so that we can cancel out based on that.1891

We have (n - 1)! on top, and (n + 1)!...well, that is going to be n + 1 times one less than that, which is going to be n,1897

times one less than that, n - 1, times one less than that...1906

Well, if we keep going down forever, that is going to be (n - 1)! here.1909

So, we have (n - 1)! on the top and n + 1 times n times (n - 1)! on the bottom.1912

Well, we can cancel the (n - 1)!'s now; and we have the limit as n goes to infinity of 1/(n + 1)(n).1918

So now, we can see that, as n goes off to infinity, well, our top doesn't change at all; it is just a constant in this case.1929

So, since our top isn't ever going to change, but our bottom, (n + 1) times n, is going to get larger and larger and larger1935

as n goes off to infinity, that means our bottom is growing, but our top is just staying the same.1940

So, in the long run, it is going to get crushed down to 0; the fraction will get crushed down to 0, so the limit of this sequence is 0.1944

All right, that finishes our exploration of limits in this course.1951

We are now going to move on to derivatives, and we will get a cool sense for how derivatives work.1955

It is really great stuff; we are getting a chance to see a preview of calculus, which is going to be really useful1958

for when we get to calculus, because we are setting a groundwork here1962

that you will then be able draw upon later, when you learn this stuff again.1964

All right, we will see you at Educator.com later--goodbye!1967