Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 7 - Exponential Functions - 7.7 L'Hôpital's Rule - Exercises - Page 366: 29



Work Step by Step

We have $$ \lim _{x \rightarrow 0} \frac{\sin x -x\cos x}{x-\sin x} =\frac{0}{0} $$ is an intermediate form, then we can apply L’Hôpital’s Rule as follows $$ \lim _{x \rightarrow 0} \frac{\sin x -x\cos x}{x-\sin x} =\lim _{x \rightarrow 0} \frac{-x\sin x}{1-\cos x} =\frac{0}{0} .$$ Again we can apply L’Hôpital’s Rule $$ \lim _{x \rightarrow 0} \frac{- \sin x-x\cos x}{-\sin x} =\frac{0}{0} .$$ Applying L’Hôpital’s Rule, we get $$ \lim _{x \rightarrow 0} \frac{- \sin x-x\cos x}{-\sin x} = \lim _{x \rightarrow 0} \frac{-2\cos x+x \sin x}{-\cos x}=2 .$$
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