Answer
$$\lim _{x \rightarrow \frac{\pi}{2}} \frac{\cot x}{\csc x}=0$$
Work Step by Step
Given $$\lim _{x \rightarrow \frac{\pi}{2}} \frac{\cot x}{\csc x}$$
let $$ f(x) = \frac{\cot x}{\csc x}$$
Since, we have
$$ f(\frac{\pi}{2})= \frac{\cot \frac{\pi}{2}}{\csc \frac{\pi}{2}}=\frac{0}{1}=0$$
So, we get
\begin{aligned}
L&= \lim _{x \rightarrow \frac{\pi}{2}} \frac{\cot x}{\csc x}\\
&= \lim _{x \rightarrow \frac{\pi}{2}} \frac{\cos x}{\sin x} \frac{1}{\csc x}\\
&= \lim _{x \rightarrow \frac{\pi}{2}} \frac{\cos x}{\sin x} \sin x \\
&= \lim _{x \rightarrow\frac{\pi}{2}} \cos x\\
&=\cos \frac{\pi}{2}\\
&=0
\end{aligned}
