Which is the solution of the limit, $\lim_{x \rightarrow \infty} x\sin (\frac{1}{x})$ Which is the solution of the limit,  
$$
\lim_{x \rightarrow \infty} x\sin \left(\frac{1}{x}\right)
$$

You can do this substitution:
$$x=\frac{1}{u},\ u=\frac{1}{x},\ x\rightarrow\infty\Rightarrow u\rightarrow0$$
Hence, your limit changes to:
$$\lim_{u\rightarrow0}\frac{1}{u}\sin(u)=\lim_{u\rightarrow0}\frac{\sin(u)}{u}$$
A very known limit.
