## Calculus (3rd Edition)

$$2$$
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 $$\lim _{x \rightarrow 0} \frac{- \sin x-x\cos x}{-\sin x} = \lim _{x \rightarrow 0} \frac{-2\cos x+x \sin x}{-\cos x}=2 .$$