Answer
$-\infty$
Work Step by Step
As $x\rightarrow 2\pi^{-}$, we note that it is positive (close to $ 2\pi$).
With the unit circle in mind,
$x$ is in quadrant IV, where the sine is negative
so $\sin x$ is negative and approaching 0
Cosecant and sine are reciprocal, so as $x\rightarrow 2\pi^{-}$,
$\displaystyle \csc x=\frac{1}{\sin x}\rightarrow-\infty$
So, the product ($x\csc x$)$\rightarrow-\infty$
