## Calculus 8th Edition

$-\infty$
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 ($x\csc x$)$\rightarrow-\infty$