Mathematics - Limits Question with Solution | TestHub

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