Explanation Procedure used: L’ Hospital’s rule. 1. Check whether the lim x → a f ( x ) g ( x ) is an indeterminate form 0 0 or ± ∞ ± ∞ . 2. If so obtain the derivative of f ( x ) and g ( x ) , separately. 3. Compute the lim x → a f ′ ( x ) g ′ ( x ) , if this limit exist then lim x → a f ( x ) g ( x ) = lim x → a f ′ ( x ) g ′ ( x ) . 4. If it does not exist repeat the step 2 and step 3. Calculation: From the given limit, f ( x ) = ( 2 e 5 x − 25 x 2 − 10 x − 2 ) and g ( x ) = ( 5 x 3 ) . Step 1: Obtain the value of lim x → 0 2 e 5 x − 25 x 2 − 10 x − 2 5 x 3 . lim x → 0 ( 2 e 5 x − 25 x 2 − 10 x − 2 5 x 3 ) = ( 2 e 5 ( 0 ) − 25 ( 0 ) 2 − 10 ( 0 ) − 2 5 ( 0 ) 3 ) = 2 − 2 0 = 0 0 This is observed to be an indeterminate form. Thus, apply L’Hospital’s rule. Step 2: Obtain the derivative of f ( x ) . f ′ ( x ) = d d x ( 2 e 5 x − 25 x 2 − 10 x − 2 ) = 10 e 5 x − 50 x − 10 Obtain the derivative of g ( x ) . g ′ ( x ) = d d x ( 5 x 3 ) = 15 x 2 Step 3: Compute the value of lim x → 0 f ′ ( x ) g ′ ( x ) . lim x → 0 f ′ ( x ) g ′ ( x ) = lim x → 0 ( 10 e 5 x − 50 x − 10 15 x 2 ) = 10 e 5 ( 0 ) − 50 ( 0 ) − 10 15 ( 0 ) 2 = 10 − 10 0 = 0 0 This is also observed to be an indeterminate form

Calculus with Applications (11th Edition)

11th Edition

ISBN: 9780321979421

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Chapter 12.7, Problem 26E

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The value of limx→02e5x−25x2−10x−25x3 by applying the L’ Hospital’s rule.

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Chapter 12.7, Problem 25E

Chapter 12.7, Problem 27E

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Calculus with Applications (11th Edition)

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The value of lim x → 0 2 e 5 x − 25 x 2 − 10 x − 2 5 x 3 by applying the L’ Hospital’s rule.

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The value of limx→02e5x−25x2−10x−25x3 by applying the L’ Hospital’s rule.

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