Answer
$$
\lim _{x \rightarrow 0} ( \sin 5 x \csc 3 x ) = \frac{5}{3}
$$
Work Step by Step
$$
\lim _{x \rightarrow 0} (\sin 5 x \csc 3 x)
$$
We can deal with it by writing the product $ \sin 5 x \csc 3 x $ as a quotient $ \frac{\sin 5 x}{\sin 3 x} $.
Since
$$
\lim _{x \rightarrow 0} ( \sin 5 x)=0
$$
and
$$
\lim _{x \rightarrow 0} ( \sin 3 x)=0
$$
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:
$$
\begin{aligned}
\lim _{x \rightarrow 0} \sin 5 x \csc 3 x & =\lim _{x \rightarrow 0} \frac{\sin 5 x}{\sin 3 x} \\
& \stackrel{\mathrm{H}}{=} \lim _{x \rightarrow 0} \frac{5 \cos 5 x}{3 \cos 3 x} \\
& =\frac{5 \cdot 1}{3 \cdot 1} \\
&=\frac{5}{3}
\end{aligned}
$$
