University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 4 - Section 4.5 - Indeterminate Forms and L'Hôpital's Rule - Exercises - Page 248: 74



Work Step by Step

L'Hospital's rule states that $\lim\limits_{x \to \infty} f(x)=\lim\limits_{x \to \infty} \dfrac{A'(x)}{B'(x)}$ Here, $\lim\limits_{x \to 0} f(x)=\lim\limits_{x \to \infty} xe^{1/x}$ and $\lim\limits_{x \to 0} \dfrac{e^{1/x}}{1/x}=\dfrac{\infty}{\infty}$ This shows an indeterminate form of the limit, so we need to use L'Hospital's rule. $\lim\limits_{x \to 0} \dfrac{e^{1/x}(-1/x^2)}{-1/x^2}=\lim\limits_{x \to 0} e^{1/x}=\infty$
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