Answer
$$1$$
Work Step by Step
Given
$$\lim _{x \rightarrow \infty}x\sin\frac{1}{x}$$
Let $t= \frac{1}{x}$ , then
\begin{aligned}
\lim _{x \rightarrow \infty}x\sin\frac{1}{x}&=\lim _{t \rightarrow 0^+}\frac{\sin t}{t} \\
&=1
\end{aligned}