Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 6 - Inverse Functions - 6.2* The Natural Logarithmic Functions - 6.2* Exercises - Page 446: 16



Work Step by Step

Find limit $\lim\limits_{x \to \infty}[ln(2+x)-ln(1+x)]$ Consider $[ln(2+x)-ln(1+x)]$ Use logarithmic property, $lnx-lny=ln\frac{x}{y}$ Thus, $[ln(2+x)-ln(1+x)]=ln[\frac{2+x}{1+x}]$ Now we will find the limit. $\lim\limits_{x \to \infty}[ln(2+x)-ln(1+x)]=\lim\limits_{x \to \infty}[ln[\frac{2+x}{1+x}]$ or $\lim\limits_{x \to \infty}[ln[\frac{2+x}{1+x}]=\lim\limits_{x \to \infty}[ln\frac{\frac{2}{x}+1}{\frac{1}{x}+1}]$ Since ${x \to \infty}$ then ${\frac{2}{x} \to \ 0}$ and ${\frac{1}{x} \to \ 0}$ Hence, $\lim\limits_{x \to \infty}[ln(2+x)-ln(1+x)]=\lim\limits_{x \to \infty}ln[\frac{0+1}{0+1}]=ln1 =0$
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