Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 7: Transcendental Functions - Section 7.5 - Indeterminate Forms and L'Hopital's Rule - Exercises 7.5 - Page 410: 74



Work Step by Step

Here, we have $\lim\limits_{x \to 0} f(x)=\lim\limits_{x \to \infty} xe^{1/x}$ But $\lim\limits_{x \to 0} \dfrac{e^{1/x}}{\frac{1}{x}}=\dfrac{\infty}{\infty}$ This shows an indeterminate form of limit, thus we will apply L-Hospital's rule such as: $\lim\limits_{x \to \infty} f(x)=\lim\limits_{x \to \infty} \dfrac{p'(x)}{q'(x)}$ $\lim\limits_{x \to 0} \dfrac{e^{1/x}(\frac{-1}{x^2})}{\frac{-1}{x^2}}=\lim\limits_{x \to 0} e^{1/x}$ or, $e^{1/0}=\infty$
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