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INSTRUCTORS Raffi Hovasapian John Zhu
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For more information, please see full course syllabus of Calculus AB
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One Sided Limits

  • Limits can differ when taken from different approaching directions
  • Limit only exists when , where a is a constant
  • In other words: limits exists at a point when the function converges at that point

One Sided Limits

By inspecting the following graph, find limx → 2 y
  • Coming from either side, y is going to 4 as x approaches 2
limx → 2 y = 4
By inspecting the graph above, find limx → 2 y
  • The limit coming from the left appears to be -2. Coming from the right it appears to be 2. In order for the limx → 2 to exist, limx → 2+ must equal limx → 2, which isn't the case here. LIMIT DOES NOT EXIST.
Limit Does Not Exist
Find limx → 0+ [2/x]
  • f(.1) = 20
  • f(.01) = 200
  • f(.0001) = 20000
limx → 0+ [2/x] = ∞
Find limx → 0 [2/x]
  • f(−.1) = −20
  • f(−.01) = −200
  • f(−.0001) = −20000
limx → 0 [2/x] = −∞
Find limx → 0 [2/x]
  • limx → 0+ [2/x] ≠ limx → 0 [2/x], therefore limx → 0 [2/x] does not exist
Limit Does Not Exist
Find limx → 4 [1/(x − 4)]
  • f(3.9) = −10
  • f(3.99) = −100
  • f(3.9999) = −10000
limx → 4 [1/(x − 4)] = −∞
Find limx → 0 [x/(|x|)]
  • Our initial reaction might be to factor to a 1, but what happens when we input a negative value?
  • f(−.01) = [(−1)/1] = −1
  • f(−200) = [(−200)/200] = −1
  • f(.01) = [1/1] = 1
  • f(200) = [200/200] = 1
  • This is a piecewise function, meaning it can be expressed as two separate functions that each have different domains. f(x) = 1 for x ≥ 0 and f(x) = -1 for x < 0. The limit as x → 0 is not equivalent to the limit as x → 0+. limx → 0 [x/(|x|)] does not exist
Limit Does Not Exist
Find:
limx → 2+ f(x)if f(x) = 3x : x ≥ 2 −x : x < 2
  • We're coming at 2 from the right side, so we only care about the function when f(x) = 3x.
  • limx → 2+ f(x) = limx → 2+ 3x = 6
6
Find
limx → 2 f(x) if f(x) = 3x : x ≥ 2 −x : x < 2
  • limx → 2 −x = −2
−2
Find limx → −3 [1/(x3 + 27)]
  • We can see that the denominator is going to approach 0 as x → −3. With a constant non-zero numerator this implies some sort of infinitely large number. Now we need to find the sign.
  • f(−3.001) = [1/(−(3 + .001)3 + 33)] < 0
  • f(−2.999) = [1/(−(3 − .001)3 + 33)] > 0
  • limx → −3 [1/(x3 + 27)] does not exist
Limit Does Not Exist

*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.

Answer

One Sided Limits

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
  • Types of Limits: One-Sided Limit Rules 0:06
    • Example
    • Applying Same Rule
    • Rule to Keep In Mind
  • Types of Limits: One-Sided Limit Example 1 1:12
    • Limit of x² From Negative Side
  • Types of Limits: One-Sided Limit, Example 2 2:27
  • Types of Limits: One-Sided Limit, Example 3 4:26
  • Types of Limits: One-Sided Limit, Example 4 5:47
  • One-Sided Limit Example: X with Even Degree Polynomial 7:00
  • One-Sided Limit Example: Entire Denominator Squared 8:09