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### L'Hopital's Rule

• Use for indeterminate forms: limits that equal
• Using L’Hôpital’s Rule:
• Check conditions
• Find
• Evaluate to get

### 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.