Explanation Given information: The function is lim x → ∞ 5 x 3 − 2 x 7 x 3 + 3 . L’Hospital’s rule: Suppose that f ( a ) = g ( a ) = 0 that f and g are differentiable on an open interval I containing a, and that g ′ ( x ) ≠ 0 on I if x ≠ a then, lim x → a f ( x ) g ( x ) = lim x → a f ′ ( x ) g ′ ( x ) Assuming that the limit on the right side of this equation exists. Calculation: Show the function as follows: lim x → ∞ 5 x 3 − 2 x 7 x 3 + 3 Substitute ∞ for x . lim x → ∞ 5 x 3 − 2 x 7 x 3 + 3 = lim x → ∞ 5 x 3 − 2 x 7 x 3 + 3 = 5 ( ∞ ) 3 − 2 ( ∞ ) 7 ( ∞ ) 3 + 3 = ∞ ∞ Consider f ( x ) = 5 x 3 − 2 x Differentiate f ( x ) with respect to x . f ′ ( x ) = 15 x 2 − 2 Consider g ( x ) = 7 x 3 + 3 Differentiate g ( x ) with respect to x . g ′ ( x ) = 21 x 2 Apply the L Hospital rule as follows: lim x → ∞ f ( x ) g ( x ) = lim x → ∞ f ′ ( x ) g ′ ( x ) (1) Substitute ( 15 x 2 − 2 ) for f ′ ( x ) and 21 x 2 for g ′ ( x ) in Equation (1)

University Calculus: Early Transcendentals (4th Edition)

4th Edition

ISBN: 9780134995540

Author: Joel R. Hass, Christopher E. Heil, Przemyslaw Bogacki, Maurice D. Weir, George B. Thomas Jr.

Publisher: PEARSON

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Chapter 4.5, Problem 11E

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Find the limit by using L’Hospital’s rule.

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Chapter 4.5, Problem 10E

Chapter 4.5, Problem 12E

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Chapter 4 Solutions

University Calculus: Early Transcendentals (4th Edition)

Ch. 4.1 - In Exercises 16, determine from the graph whether...Ch. 4.1 - In Exercises 1–6, determine from the graph whether...Ch. 4.1 - Prob. 3E Ch. 4.1 - In Exercises 1–6, determine from the graph whether...Ch. 4.1 - Prob. 5E Ch. 4.1 - Prob. 6E Ch. 4.1 - In Exercises 7–10, find the absolute extreme...Ch. 4.1 - In Exercises 7–10, find the absolute extreme...Ch. 4.1 - Prob. 9E Ch. 4.1 - Prob. 10E

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