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. Given: The given limit is lim x → 0 ( e 3 x − 9 2 x 2 − 3 x − 1 x 3 ) . Calculation: Step 1: Obtain the value of lim x → 0 ( e 3 x − 9 2 x 2 − 3 x − 1 x 3 ) . lim x → 0 ( e 3 x − 9 2 x 2 − 3 x − 1 x 3 ) = e 3 ( 0 ) − 9 2 ( 0 ) 2 − 3 ( 0 ) − 1 0 3 = 1 − 0 − 0 − 1 0 = 0 0 Observe the solution is in an indeterminate form, apply L’Hospital’s rule. Step 2: From given, f ( x ) = e 3 x − 9 2 x 2 − 3 x − 1 and g ( x ) = x 3 Obtain the derivative of f ( x ) f ′ ( x ) = d d x ( e 3 x − 9 2 x 2 − 3 x − 1 ) = 3 e 3 x − 9 x − 3 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 ) . lim x → 0 f ′ ( x ) g ′ ( x ) = lim x → 0 ( 3 e 3 x − 9 x − 3 3 x 2 ) = 3 e 3 ( 0 ) − 9 ( 0 ) − 3 3 ( 0 ) 2 = 3 − 0 − 3 0 = 0 0 Observe the solution is in an indeterminate form, apply L’Hospital’s rule

Calculus With Applications, Books a la Carte Plus MyLab Math Package (11th Edition)

11th Edition

ISBN: 9780133886849

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

Publisher: PEARSON

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Chapter 12.7, Problem 4YT

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Chapter 12 Solutions

Calculus With Applications, Books a la Carte Plus MyLab Math Package (11th Edition)

Ch. 12.1 - Find the first four terms of the sequence having...Ch. 12.1 - Prob. 2YT Ch. 12.1 - Prob. 3YT Ch. 12.1 - Prob. 4YT Ch. 12.1 - Prob. 5YT Ch. 12.1 - Prob. 1E Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E Ch. 12.1 - List the first n terms of the geometric sequence...Ch. 12.1 - Prob. 5E

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