Answer
The limit does not exist.
Work Step by Step
Our aim is to evaluate the limit for $\lim\limits_{x \to 0^{+}} \dfrac{x \sin (1/x)}{\sin x}$
or, $=\lim\limits_{x \to 0^{+}} \dfrac{x }{\sin x} \times \lim\limits_{x \to 0^{+}} \sin (\dfrac{1}{x})$
or, $=(1) \times \lim\limits_{x \to 0^{+}} \sin (\dfrac{1}{x})$
But $\lim\limits_{x \to 0^{+}} \sin (\dfrac{1}{x})$ shows a finite but not a fixed quantity between $1$ and $-1$. This means that the limit for the given function does not exist.
