L' Hospitals Notes

# L' Hospitals Notes - – Try factoring which will usually...

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L’Hospital’s Rule Evaluate the following limits: lim x 2 x - 2 x - 3 lim x 3 x - 3 x 2 - 9 lim x 0 sin x x If f and g are diﬀerentiable functions and g 0 ( x ) 6 = 0 near a and: lim x a f ( x ) = 0 and lim x a g ( x ) = 0 OR lim x a f ( x ) = ±∞ and lim x a g ( x ) = ±∞ then lim x a f ( x ) g ( x ) = lim x a f 0 ( x ) g 0 ( x ) CAREFUL! You MUST have 0 0 or to use L’Hospital’s Rule. Indeterminate Forms: 0 0 , 0 ·∞ • ∞-∞ 0 0 , 0 , 1 The following are NOT indeterminate forms: + ∞ → ∞ -∞-∞ → -∞ 0 0 0 -∞ → ∞

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Indeterminate Forms: 0 0 , Example 1. Evaluate lim x 3 x - 3 x 2 - 9 Example 2. Evaluate lim x 0 e x - 1 sin x Example 3. Evaluate lim x 0 x tan - 1 (4 x ) Example 4. Evaluate lim y →∞ y 2 ln y Example 5. Evaluate lim t 0 1 - e 3 t + 3 t t 2
Indeterminate Forms: 0 ·∞ , ∞-∞ The goal is to change the expression so that it has the form 0 0 or . Common tricks include: For 0 ·∞ : The function will have the form f ( x ) · g ( x ). Rewrite as f ( x ) 1 g ( x ) or g ( x ) 1 f ( x ) If trig functions are present, try converting to sin x and cos x . For ∞-∞ : If the expression is rational (looks like fractions) ﬁnd a common denomi- nator and subtract to obtain a single expression.

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Unformatted text preview: – Try factoring, which will usually get you something of the form 0 ·∞ Example 6. Evaluate lim θ → 0+ θ cot θ Example 7. Evaluate lim x → 0+ ( e x-1) ln x Example 8. Evaluate lim x → 1 ± 1 e x-e-1 x 2-1 ² Example 9. Evaluate lim x → π 2-(sec x-tan x ) Indeterminate Forms: , ∞ , 1 ∞ First, we must recall that lim x → a f ( x ) g ( x ) = lim x → a e ln f ( x ) g ( x ) = e lim x → a ln f ( x ) g ( x ) = e lim x → a g ( x ) ln f ( x ) We will use this fact to rewrite our functions to the indeterminate form 0 ·∞ and then use the above methods to evaluate the limit. Example 10. Evaluate lim x → 0+ x 3 x Example 11. Evaluate lim x → (cos x ) x Example 12. Evaluate lim x → 0+ (-ln x ) x...
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## This note was uploaded on 10/02/2011 for the course MATH 2214 at Virginia Tech.

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L' Hospitals Notes - – Try factoring which will usually...

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