For more information, please see full course syllabus of College Calculus: Level I

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For more information, please see full course syllabus of College Calculus: Level I

For more information, please see full course syllabus of College Calculus: Level I

### L'Hopital's Rule

- Start by checking to see if you are dealing with an indeterminate type of limit of the form or . If so, proceed with L’Hopital’s Rule.
- Remember that it may be necessary to use L’Hopital’s Rule more than once!
- If your limit is of another indeterminate type such as , , , , or , then you can rearrange the problem into a L’Hopital’s Rule form.
- If your limit is not indeterminate at all, then simply complete the limit computation by ordinary methods!

### L'Hopital's Rule

Given lim

_{x → π}[cosx/3x], can L'Hopital's Rule be applied?- Identify conditions
- f(π) = − 1
- g(π) = 3π

It cannot because f(π) ≠ g(π) ≠ 0

Given lim

_{x → π}[x/sinx], can L'Hopital's Rule be applied?- Identify conditions
- f(0) = 0
- g(0) = 0
- f′(0) = 1
- g′(0) = cos0
- = 1

Yes, L'Hopital's Rule can be applied.

Given lim

_{x → 0}[(√{8 − x} − 8)/x], can L'Hopital's Rule be applied? If so, what is the limit?- Identify conditions
- f(0) = √{8 − 0} − 8
- = 2√2 − 8

It cannot be applied because f(0) ≠ (0) ≠ 0

Given lim

_{θ→ π}[(tanθ)/(3cosθ+ 3)], can L'Hopital's Rule be applied? If so, what is the limit?- Identify conditions
- g′(π) = − 3sinπ
- = 0

It cannot be applied because g′(π) = 0

Given lim

_{x → ∞}[(x + 4)/(2 − 5x)], can L'Hopital's Rule be applied? If so, what is the limit?- Identify conditions
- f(∞) = ∞+ 4
- = ∞
- g(∞) = 2 − 5(∞)
- = ∞
- f′(x) = 1
- g′(x) = − 5
- Apply L'Hopital's Rule

lim

_{x → ∞}[(x + 4)/(2 − 5x)] = [1/( − 5)]Given lim

_{y → − 2}[(7y + 14)/(3y^{2}− 12)], can L'Hopital's Rule be applied? If so, what is the limit?- Identify conditions
- f( − 2) = 7y + 14
- = 7( − 2) + 14
- = 0
- g( − 2) = 3y
^{2}− 12 - = 3( − 2)
^{2}− 12 - = 3(4) − 12
- = 0
- f′( − 2) = 7
- g′( − 2) = 6( − 2)
- = − 12

Apply L'Hopital's Rule

lim

lim

_{y − 2}[(7y + 14)/(3y^{2}− 12)] = [7/(−12)]Given lim

_{b → ∞}[(b^{3}+ 8b^{2})/3], can L'Hopital's Rule be applied? If so, what is the limit?- Identify conditions

- g(∞) = 3

It cannot be applied because g(∞) ≠ 0 or ∞

Find lim

_{x → ∞}[(3^{x})/(9x + 2)]- Identify conditions
- f(∞) = 3
^{∞} - = ∞
- g(∞) = 9x + 2
- = ∞
- f′(∞) = (ln3)3
^{∞} - g′(x) = 9
- Apply L'Hopital's Rule

lim

_{x → ∞}[(3^{x})/(9x + 2)] = [((ln3)3^{∞})/9]Find lim

_{x → 0}[(6x^{2})/(e^{x}− 1)]- Identify conditions
- f(0) = 6x
^{2} - = 6(9)
^{2} - = 0
- g(0) = e
^{x}− 1 - = e
^{0}− 1 - = 1 − 1
- = 0
- f′(0) = 12(0)
- = 0
- g′(0) = e
^{0} - = 1

Apply L'Hopital's Rule

lim

lim

_{y → − 2}[(6x^{2})/(e^{x}− 1)] = [0/1] = 0Given lim

_{x → ∞}[(3x^{1/3})/(e^{x}(x^{2}+ 1))], can L'Hopital's Rule be applied? If so, what is the limit?- Identify conditions
- f(∞) = 3(∞)
^{1/3} - = ∞
- g(∞) = e
^{∞}(∞^{2}+ 1) - = ∞
- f′(∞) = x
^{ − 2/3} - = ∞
^{ − 2/3} - = 0
- g′(∞) = e
^{x}(2x) + e^{x}(x^{2}) - = e
^{∞}(2(∞)) + e^{∞}(∞^{2}) - = ∞
- Apply L'Hopital's Rule
- lim
_{x∞}[(3x^{1/3})/(e^{x}(x^{2}+ 1))] = [0/(∞)]

= 0

*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

### L'Hopital's Rule

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
- Using L'Hopital's Rule 0:09
- Informal Definition
- Lecture Example 1 1:27
- Lecture Example 2 4:00
- Lecture Example 3 5:40
- Lecture Example 4 9:38
- Additional Example 5
- Additional Example 6

0 answers

Post by ansam alfaouri on April 3, 2014

I have a question how to find two functions such as that the limit as x approaches infinity f(x)=infinity and as x aproches the infinity g(x)=infinity

0 answers

Post by Constantin Ficiu on November 11, 2013

Great examples and very well explained.

Thank you.

0 answers

Post by Stephanie Sergent on June 27, 2012

could you explain by braking it up into smaller steps?

0 answers

Post by Stephanie Sergent on June 27, 2012

example 6 was too complicated.

1 answer

Last reply by: amera abdo

Mon Jan 2, 2012 5:04 PM

Post by amera abdo on January 2, 2012

What happens to the -1 at the fourth step? how did u get rid of it?