Answer
$$
\lim _{x \rightarrow \infty} x \sin (\pi / x) =\pi
$$
Work Step by Step
$$
\lim _{x \rightarrow \infty} x \sin (\pi / x)
$$
We can deal with it by writing the product $ x \sin (\pi / x) $as a quotient $\frac{\sin (\pi / x)}{1 / x} $.
Since
$$
\lim _{x \rightarrow \infty} ( \sin (\pi / x))=0
$$
and
$$
\lim _{x \rightarrow \infty } ( 1 / x)=0
$$
So, we find that, this convert the given limit into an indeterminate form of type $\frac{0}{0}$ and we can apply l’Hospital’s Rule:
$$
\begin{aligned}
\lim _{x \rightarrow \infty} x \sin (\pi / x) &=\lim _{x \rightarrow \infty} \frac{\sin (\pi / x)}{1 / x} \\ \stackrel{\mathrm{H}}{=}
& \lim _{x \rightarrow \infty} \frac{\cos (\pi / x)\left(-\pi / x^{2}\right)}{-1 / x^{2}} \\
&=\pi \lim _{x \rightarrow \infty} \cos (\pi / x) \\
&=\pi(1)\\
&=\pi
\end{aligned}
$$
