_ x arrow 0 ((x- x)/(x)) ((1)/(x))

Consider $\lim _{x \rightarrow 0}\left(\frac{x-\sin x}{x}\right) \sin \left(\frac{1}{x}\right)$=\lim _{x \rightarrow 0}\left[\frac{x\left(1-\frac{\sin x}{x}\right)}{x}\right] \times \lim _{x \rightarrow 0} \sin \left(\frac{1}{x}\right)$=\lim _{x \rightarrow 0}\left[1-\frac{\sin x}{x}\right] \times \lim _{x \rightarrow 0} \sin$\left(\frac{1}{x}\right)$=\left[1-\lim _{x \rightarrow 0} \frac{\sin x}{x}\right] \times \lim _{x \rightarrow 0} \sin \left(\frac{1}{x}\right)$=0 \times \lim _{x \rightarrow 0} \sin \left(\frac{1}{x}\right)=0