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Unformatted text preview: L’Hospital’s Rule John E. Gilbert, Heather Van Ligten, and Benni Goetz After getting used to the idea of what a LIMIT really means, you learned how to decide if lim x → 1 x- 1 x 2- 1 , lim x → sin x + x tan x , lim x → parenleftBigg 1 x- 1 x ( x + 1) parenrightBigg exist and to find the value of their limit. Usually algebra along with Properties of Limits or standard limits like lim x → sin x x = 1 was used. But what if the limit had been lim x → 1 ln x x 2- 1 ? Direct evaluation wouldn’t have worked because if you’d set f ( x ) = ln x and g ( x ) = x 2- 1 , then lim x → 1 ln x x 2- 1 = lim x → 1 f ( x ) g ( x ) = , which wouldn’t have helped since is an Indeterminate Form , meaning that it’s value can’t be determined. We are about to see that differentiation provides another option - after all, this is calculus! L’Hospital’s Rule: if f, g are differentiable on an open in- terval containing a , except possibly at x = a , and if lim x → a f ( x ) g ( x ) = , then lim x → a f ( x ) g ( x ) = lim x → a f ′ ( x ) g ′ ( x ) whenever this last limit exists. In the earlier case when f ( x ) = ln x and g ( x ) = x 2- 1 , then a = 1 and L’Hospital says lim x → 1 parenleftBigg f ( x ) g ( x ) parenrightBigg = lim x → 1 f ′ ( x ) g ′ ( x ) = lim x → 1 1 /x 2 x = 1 2 . Simple, right? It gets better! L’Hospital’s Rule works also for one-sided limits: replace x → a everywhere with one of x → a − , x → a + , x → ∞ , x → -∞ ....
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This note was uploaded on 04/12/2011 for the course M 408d taught by Professor Sadler during the Spring '07 term at University of Texas.
- Spring '07
- Multivariable Calculus