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: 31


$\ln 2$

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, $\lim\limits_{x \to \infty} f(x)=\lim\limits_{x \to \infty} \dfrac{\ln (x+1)}{\log_2 x}=\dfrac{\infty}{\infty}$ This shows an indeterminate form of a limit, so we need to use L'Hospital's rule. $ \lim\limits_{x \to \infty} \dfrac{1/x+1}{1/x \ln2}=\ln 2 \lim\limits_{x \to \infty} \dfrac{x}{x+1}=\dfrac{\infty}{\infty}$ Again apply L'Hospital's rule. $\lim\limits_{x \to \infty} f(x)=(\ln 2) \lim\limits_{x \to \infty} \dfrac{1}{1}=\ln 2$
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