Explanation Given: The provided function is lim x → ∞ x 3 e 2 x . Also, function f ( x ) = x 3 , g ( x ) = e 2 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 3 e 2 x . Apply the limit x → ∞ to the function x 3 e 2 x . Substitute ∞ for x . lim x → ∞ x 3 e 2 x = ( ∞ ) 3 e 2 ⋅ ∞ = ∞ ∞ 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 3 e 2 x = lim x → ∞ d d x ( x 3 ) d d x ( e 2 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 3 e 2 x = lim x → ∞ d d x ( x 3 ) d d x ( e 2 x ) = lim x → ∞ 3 x 3 − 1 2 e 2 x = lim x → ∞ 3 x 2 2 e 2 x Now, again apply the limit x → ∞ . Substitute ∞ for x . lim x → ∞ x 3 e 2 x = lim x → ∞ 3 x 2 2 e 2 x = 3 ∞ 2 2 e 2 ⋅ ∞ = ∞ ∞ As the direct substitution of limit produces indeterminate form ∞ / ∞ , so L’Hospital’s Rule can be applied

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Chapter 8.7, Problem 70E

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To calculate: The comparative rates of increase of the function limx→∞x3e2x by the use of L’Hospital’s Rule.

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Chapter 8.7, Problem 69E

Chapter 8.7, Problem 71E

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The comparative rates of increase of the function lim x → ∞ x 3 e 2 x by the use of L’Hospital’s Rule.

Solution Summary: The author calculates the comparative rates of increase of the function undersetxto.

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To calculate: The comparative rates of increase of the function limx→∞x3e2x by the use of L’Hospital’s Rule.

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