Answer
$\dfrac{1}{2}$
Work Step by Step
Consider: $\lim\limits_{x \to \dfrac{\pi}{2}}f(x)=\lim\limits_{x \to \dfrac{\pi}{2}}\dfrac{\ln (\csc)}{(x-(\dfrac{\pi}{2}))^2}$
We need to check that the limit has an indeterminate form.
Thus, $f(\dfrac{\pi}{2})=\dfrac{0}{0}$
The limit shows an indeterminate form. Thus, apply L-Hospital's rule: $\lim\limits_{a \to b}f(x)=\lim\limits_{a \to b}\dfrac{g'(x)}{h'(x)}$
Then
$\lim\limits_{x \to \dfrac{\pi}{2}}\dfrac{-\cot x}{2x-\pi}=\dfrac{0}{0}$
Need to apply L-Hospital's rule again.
$\lim\limits_{x \to \dfrac{\pi}{2}}\dfrac{\csc^2 \dfrac{\pi}{2}}{2}=\dfrac{1}{2}$