calculus - Solve limit containing (1/0) using L'Hopital's Rule

I was assigned a problem in my calculus 2 course that I can't seem to solve. We're going over indeterminate forms solved by L'Hopital's Rule.

$$\lim_{x\to \pi^-} \left( \frac{\csc(x)+x}{\tan(\frac{x}{2})} \right)$$

This is in the indeterminate form $\left[\frac{\infty}{\infty}\right]$, which is given, except I'm not sure why, as $\csc(\pi)$ is undefined. Is this because trig functions occur infinitely (within their range), e.g. $\sin(x) = 1$, occurs for infinite $x$'s?

After applying L'Hopital's Rule, I get:

$$\lim_{x\to \pi^-} \left( \frac{-\csc(x)\cot(x) + 1}{\frac{1}{2}\sec^2(\frac{x}{2})} \right)$$

However, I don't understand how I'm supposed to solve from here. No matter how many times we derive, $\csc(x)$ won't go away and I'm not aware of any trigonometric identities that can simplify the undefined division away. Am I missing something obvious?