Answer
$\infty$
Work Step by Step
Given
$$\lim _{x \rightarrow \infty} x^{3 / 2} \sin (1 / x)$$
Rewrite the limit as the following
\begin{aligned}
\lim _{x \rightarrow \infty} x^{3 / 2} \sin (1 / x)&= \lim _{x\to \infty \:}\left(\frac{\sin \left(\frac{1}{x}\right)}{\frac{1}{x^{\frac{3}{2}}}}\right)\\
\end{aligned}
Apply L'hopital's rule
\begin{aligned}
\lim _{x \rightarrow \infty} x^{3 / 2} \sin (1 / x)&= \lim _{x\to \infty \:}\left(\frac{\sin \left(\frac{1}{x}\right)}{\frac{1}{x^{\frac{3}{2}}}}\right)\\
&= \lim _{x\to \infty \:}\left(\frac{-\frac{\cos \left(\frac{1}{x}\right)}{x^2}}{-\frac{3}{2x^{\frac{5}{2}}}}\right)\\
&= \lim _{x\to \infty \:}\left(\frac{2}{3}x^{\frac{1}{2}}\cos \left(\frac{1}{x}\right)\right)\\
&= \frac{2}{3}\cdot \lim _{x\to \infty \:}\left(x^{\frac{1}{2}}\cos \left(\frac{1}{x}\right)\right)\\
&= \frac{2}{3}\cdot \lim _{x\to \infty \:}\left(x^{\frac{1}{2}}\right)\cdot \lim _{x\to \infty \:}\left(\cos \left(\frac{1}{x}\right)\right)\\
&= \frac{2}{3}\cdot \infty \cdot \:1\\
&=\infty
\end{aligned}
