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: 55



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, $\ln f(x)=(\ln x) ^{-1/\ln x} \implies\ln f(x)= \dfrac{-\ln x}{\ln x}$ or, $\ln f(x)=\frac{-(\ln x)}{\ln x} \implies f(x)=-1$ Now, $e^{\lim\limits_{x \to 0^{+}} (-1)}=e^{-1}=\dfrac{1}{e}$
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