Explanation Given: The provided function is lim x → ∞ x 2 e 5 x . Also, function f ( x ) = x 2 , g ( x ) = e 5 x . 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 Differentiation formula: d d x ( e n x ) = n e n x . Calculation: Consider the function x 2 e 5 x . Apply the limit x → ∞ to the function x 2 e 5 x . Substitute ∞ for x . lim x → ∞ x 2 e 5 x = ( ∞ ) 2 e 5 ⋅ ∞ = ∞ ∞ As the direct substitution of limit produces indeterminate form ∞ / ∞ , 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 → ∞ x 2 e 5 x = lim x → ∞ d d x ( x 2 ) d d x ( e 5 x ) Recall the power rule, d d x x n = n x n − 1 And the differentiation formula, d d x ( e n x ) = n e n x Apply the power rule and differentiation formula, lim x → ∞ x 2 e 5 x = lim x → ∞ d d x ( x 2 ) d d x ( e 5 x ) = lim x → ∞ 2 x 5 e 5 x Now, again apply the limit x → ∞

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ISBN: 9781285057095

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Chapter 8.7, Problem 69E

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To calculate: The comparative rates of increase of the function limx→∞x2e5x by the use of L’Hospital’s Rule.

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The comparative rates of increase of the function lim x → ∞ x 2 e 5 x by the use of L’Hospital’s Rule.

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To calculate: The comparative rates of increase of the function limx→∞x2e5x by the use of L’Hospital’s Rule.

Chapter 8 Solutions