Single Variable Calculus, Chapter 2, 2.2, Section 2.2, Problem 31

Find the infinite limit. $\lim\limits_{x \rightarrow 2\pi^-} x \csc x$


$
\begin{array}{|c|c|}
\hline
x & f(x)\\
\hline
2\pi - 0.1 & -61.9350\\
2\pi - 0.01 & -627.3289\\
2\pi - 0.001 & -6282.1864\\
2\pi - 0.0001 & -62830.8532\\
\hline
\end{array}
$


As based from the table, as $x$ approaches $2\pi$ from the left, the value of the limit approaches $-\infty$


$
\begin{equation}
\begin{aligned}
\lim\limits_{x \rightarrow 2\pi^-} x \csc x & = \lim\limits_{x \rightarrow 2\pi^-} \displaystyle \frac{x}{\sin x} = \frac{2 \pi - 0.0001}{\sin (2 \pi - 0.001)}\\
\lim\limits_{x \rightarrow 2\pi^-} x \csc x & = -62830.8532 \\
\lim\limits_{x \rightarrow 2\pi^-} x \csc x & = - \infty
\end{aligned}
\end{equation}
$