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: 53

Answer

$1$

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, $\ln f(x)=\frac{1}{x}\ln (\ln x) \implies f(x)=e^{( \frac{\ln(\ln x)}{x})}$ Now, $e^{\lim\limits_{x \to \infty} ( \frac{\ln(\ln x)}{x})}=\dfrac{\infty}{\infty}$ This shows an indeterminate form of the limit, so we need to use L'Hospital's rule. $e^{\lim\limits_{x \to \infty} ( \frac{1/x \ln x}{1})}=e^{0}=1$
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.