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 409: 31


$\ln 2$

Work Step by Step

Consider: $\lim\limits_{x \to \infty} f(x)=\lim\limits_{x \to \infty} \dfrac{\ln (x+1)}{\log_2 x}$ and $\lim\limits_{x \to \infty} f(\infty)=\dfrac{\infty}{\infty}$ This shows an indeterminate form of limit so 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 \infty} \dfrac{1/x+1}{1/(x \ln2)}=(\ln 2) \lim\limits_{x \to \infty} \dfrac{x}{x+1}=\dfrac{\infty}{\infty}$ Now, again apply L-Hospital's rule. $(\ln 2) \lim\limits_{x \to \infty} (\dfrac{1}{1})=\ln 2$
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