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: 35


$$\lim_{x\to0}\frac{\ln(1+x)}{\cos x+e^x-1}=0$$

Work Step by Step

$$A=\lim_{x\to0}\frac{\ln(1+x)}{\cos x+e^x-1}$$ As $x$ approaches $0$, $\ln(1+x)$ approaches $\ln(1+0)=\ln1=0$ and $(\cos x+e^x-1)$ approaches $\cos0+e^0-1=1+1-1=1$ So this is not an indeterminate form, but another with which we can replace $x=0$ right away. $$A=\frac{\ln(1+0)}{\cos0+e^0-1}$$ $$A=\frac{0}{1}$$ $$A=0$$
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