Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 4 - Section 4.4 - Indeterminate Forms and l''Hospital''s Rule - 4.4 Exercises - Page 312: 62

Answer

$\lim\limits_{x \to \infty}x^{(ln~2)/(1+ln~x)}=2$

Work Step by Step

$\lim\limits_{x \to \infty}x^{(ln~2)/(1+ln~x)}=\lim\limits_{x \to \infty}(e^{ln~x})^{(ln~2)/(1+ln~x)}=\lim\limits_{x \to \infty}e^{(ln~x)(ln~2)/(1+ln~x)}$ $\lim\limits_{x \to \infty}\frac{(ln~x)(ln~2)}{1+ln~x} = \lim\limits_{x \to \infty}\frac{(ln~2/x)}{(1/x)} = ln~2$ Therefore: $\lim\limits_{x \to \infty}e^{(ln~x)(ln~2)/(1+ln~x)} = e^{ln~2} = 2$
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