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 0 answersPost by ansam alfaouri on April 3, 2014I 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 answersPost by Constantin Ficiu on November 11, 2013Great examples and very well explained.Thank you. 0 answersPost by Stephanie Sergent on June 27, 2012could you explain by braking it up into smaller steps? 0 answersPost by Stephanie Sergent on June 27, 2012example 6 was too complicated. 1 answerLast reply by: amera abdoMon Jan 2, 2012 5:04 PMPost by amera abdo on January 2, 2012What happens to the -1 at the fourth step? how did u get rid of it?

### 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 limx → 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 limx → 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 limx → 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 limθ→ 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 limx → ∞ [(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
limx → ∞ [(x + 4)/(2 − 5x)] = [1/( − 5)]
Given limy → − 2 [(7y + 14)/(3y2 − 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) = 3y2 − 12
• = 3( − 2)2 − 12
• = 3(4) − 12
• = 0
• f′( − 2) = 7
• g′( − 2) = 6( − 2)
• = − 12
Apply L'Hopital's Rule
limy − 2 [(7y + 14)/(3y2 − 12)] = [7/(−12)]
Given limb → ∞ [(b3 + 8b2)/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 limx → ∞ [(3x)/(9x + 2)]
• Identify conditions
• f(∞) = 3
• = ∞
• g(∞) = 9x + 2
• = ∞
• f′(∞) = (ln3)3
• g′(x) = 9
• Apply L'Hopital's Rule
limx → ∞ [(3x)/(9x + 2)] = [((ln3)3 )/9]
Find limx → 0 [(6x2)/(ex − 1)]
• Identify conditions
• f(0) = 6x2
• = 6(9)2
• = 0
• g(0) = ex − 1
• = e0 − 1
• = 1 − 1
• = 0
• f′(0) = 12(0)
• = 0
• g′(0) = e0
• = 1
Apply L'Hopital's Rule
limy → − 2 [(6x2)/(ex − 1)] = [0/1] = 0
Given limx → ∞ [(3x1/3)/(ex(x2 + 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′(∞) = ex(2x) + ex(x2)
• = e(2(∞)) + e(∞2)
• = ∞
• Apply L'Hopital's Rule
• limx∞ [(3x1/3)/(ex(x2 + 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.

### 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