Explanation Given information: The function is lim x → 0 + x 2 ln x . Indeterminate powers If lim x → a ln f ( x ) = L , then lim x → a f ( x ) = lim x → a e ln f ( x ) = e L Here, a may be either finite or infinite. 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 → 0 + x 2 ln x Substitute 0 for x . lim x → 0 + x 2 ln x = lim x → 0 + x 2 ln x = lim x → 0 + 0 ln 0 Consider the function f ( x ) = ln x Differentiate f ( x ) with respect to x . f ′ ( x ) = d d x ( ln x ) = 1 x Consider g ( x ) = 1 x 2 Differentiate g ( x ) with respect to x . g ′ ( x ) = d d x ( 1 x 2 ) = d d x ( x − 2 ) = − 2 x − 2 − 1 = − 2 x 3 Apply the L Hospital rule as follows: lim x → 0 + f ( x ) g ( x ) = lim x → 0 + f ′ ( x ) g ′ ( x ) (1) Substitute 1 x for f ′ ( x ) and ( − 2 x 3 ) for g ′ ( x ) in Equation (1)

University Calculus: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780321999580

Author: Joel R. Hass, Maurice D. Weir, George B. Thomas Jr.

Publisher: PEARSON

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topic…

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Chapter 4.5, Problem 63E

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Calculate the limit by using indeterminate powers.

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Chapter 4.5, Problem 62E

Chapter 4.5, Problem 64E

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Chapter 4 Solutions

University Calculus: Early Transcendentals (3rd 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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