Explanation Given: The provided function is lim x → 1 ln x 3 x 2 − 1 . Formula used: L’Hospital’s Rule: If the limit of f ( x ) / g ( x ) when x approaches c from right produces indeterminate form 0 / 0 , then lim x → c f ( x ) g ( x ) = lim x → c f ′ ( x ) g ′ ( x ) . Power rule: d d x x n = n x n − 1 Chain rule: d y d x = d y d u ⋅ d u d x Differentiation formula: d d x ( ln x ) = 1 x Calculation: Consider the function ln x 3 x 2 − 1 . Apply the limit x → 1 to the function ln x 3 x 2 − 1 . Substitute 1 for x . lim x → 1 ln x 3 x 2 − 1 = ln 1 3 1 2 − 1 = ln 1 1 − 1 = 0 0 As the direct substitution of limit produces indeterminate form 0 / 0 , so L’Hospital’s Rule can be applied. Recall the L’Hospital’s Rule, lim x → c f ( x ) g ( x ) = lim x → c f ′ ( x ) g ′ ( x ) Apply the L’Hospital’s Rule, lim x → 1 ln x 3 x 2 − 1 = lim x → 1 d d x

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Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Chapter 8.7, Problem 16E

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To calculate: The value of limit limx→1lnx3x2−1 by the use of L’Hospital’s Rule, if necessary.

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Chapter 8.7, Problem 15E

Chapter 8.7, Problem 17E

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The value of limit lim x → 1 ln x 3 x 2 − 1 by the use of L’Hospital’s Rule, if necessary.

Solution Summary: The author explains how to calculate the value of limit undersetxto 1mathrmlim

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To calculate: The value of limit limx→1lnx3x2−1 by the use of L’Hospital’s Rule, if necessary.

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Chapter 8 Solutions