7.8 - Calculus II-Stewart Dr. Berg Spring 2010 7.8...

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Calculus II- Stewart Dr. Berg Spring 2010 Page 1 7.8 7.8 L ʼ Hospital ʼ s Rule We sometimes find ourselves trying to evaluate limits like lim x →∞ ln x x , lim x 0 e x 1 x , or lim x π / 2 ( ) sec x tan x ( ) . These are known as indeterminate forms (of types , 0 0 , and ∞−∞ ) and can be evaluated using a theorem called L’Hospital’s Rule. The Indeterminate Form of Type 0/0 Theorem Suppose f and g are differentiable and g ( x ) 0 near a (except possibly at a ). If lim x a f ( x ) = 0 and lim x a g ( x ) = 0 , then lim x a f ( x ) g ( x ) = lim x a f ( x ) g ( x ) . Example A Calculate lim x 0 e x 1 x . Solution : Indeed, lim x 0 e x 1 x = lim x 0 d dx e x 1 ( ) d dx x ( ) = lim x 0 e x 1 = 1 . Example B Calculate lim x 1 2 x 2 3 x + 1 x 2 + x 2 . Solution : First note that this quotient is of the form 0/0 (it satisfies the premises of the theorem). Then lim x 1 2 x 2 3 x + 1 x 2 + x 2 = lim x 1 d dx 2 x 2 3 x + 1 ( ) d dx x 2 + x 2 ( ) = lim x 1 4 x 3 2 x + 1 = 1 3 . Alternatively lim x 1 2 x 2 3 x + 1 x 2 + x 2 = lim x 1 2 x 1 ( ) x 1 ( ) x 1 ( ) x + 2 ( ) = lim
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This note was uploaded on 05/04/2010 for the course MATH 408L taught by Professor Gogolev during the Spring '09 term at University of Texas.

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7.8 - Calculus II-Stewart Dr. Berg Spring 2010 7.8...

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