Answer
$$0$$
Work Step by Step
Since
\begin{align*}
\lim _{x \rightarrow \infty} \frac{\sin x}{\lfloor x\rfloor}&\leq \lim _{x \rightarrow \infty} \frac{1}{\lfloor x\rfloor}\\
&=0
\end{align*}
$\text { since }\lfloor x\rfloor \rightarrow \infty \text { as } x \rightarrow \infty \Rightarrow \lim _{x \rightarrow \infty} \frac{\sin x}{\lfloor x\rfloor}=0$
