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: Compute the common denominator, ( 12 e x x 3 − 12 x 3 − 12 x 2 − 6 x ) = ( 12 e x x 3 − 12 x 3 − 12 x 2 ( x x ) − 6 x ( x 2 x 2 ) ) = 12 e x x 3 − 12 x 3 − 12 x x 3 − 6 x 2 x 3 = 12 e x − 12 − 12 x − 6 x 2 x 3 Thus, rewrite the given limit as lim x → 0 ( 12 e x − 12 − 12 x − 6 x 2 x 3 ) . Step 1: Obtain the value of lim x → 0 ( 12 e x − 12 − 12 x − 6 x 2 x 3 ) . lim x → 0 ( 12 e x − 12 − 12 x − 6 x 2 x 3 ) = ( 12 e 0 − 12 − 12 ( 0 ) − 6 ( 0 ) 2 ( 0 ) 3 ) = 12 − 12 − 0 − 0 0 = 0 0 It is observed to be an indeterminate form. Thus, apply L’Hospital’s rule. Step 2: It is noticed that f ( x ) = ( 12 e x − 12 − 12 x − 6 x 2 ) and g ( x ) = x 3 . Obtain the derivative of f ( x ) . f ′ ( x ) = d d x ( 12 e x − 12 − 12 x − 6 x 2 ) = 12 e x − 12 − 12 x = 12 ( e x − 1 − x ) Obtain the derivative of g ( x ) . g ′ ( x ) = d d x ( x 3 ) = 3 x 2 Step 3: Compute the value of lim x → 0 f ′ ( x ) g ′ ( x )

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 44E

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The value of limx→0(12exx3−12x3−12x2−6x) after computing a common denominator.

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Chapter 12.7, Problem 43E

Chapter 12.7, Problem 45E

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Calculus with Applications (11th Edition)

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The value of lim x → 0 ( 12 e x x 3 − 12 x 3 − 12 x 2 − 6 x ) after computing a common denominator.

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The value of limx→0(12exx3−12x3−12x2−6x) after computing a common denominator.

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