Answer
$\lim\limits_{x \to \pi^-}(x-\pi)~csc~x = -1$
Work Step by Step
$\lim\limits_{x \to \pi^-}(x-\pi)~csc~x = \lim\limits_{x \to \pi^-}\frac{x-\pi}{sin~x}= \frac{0}{0}$
We can use L'Hospital's Rule:
$\lim\limits_{x \to \pi^-}\frac{x-\pi}{sin~x}$
$=\lim\limits_{x \to \pi^-}\frac{1}{cos~x}$
$=\frac{1}{-1}$
$= -1$
