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