To evaluate the limit

$\lim_{x \to 0} x \csc x$

we start by rewriting it in terms of sine, as $\csc x = \frac{1}{\sin x}$:

$\lim_{x \to 0} x \csc x = \lim_{x \to 0} \frac{x}{\sin x}.$

Now, we can use the standard limit result:

$\lim_{x \to 0} \frac{x}{\sin x} = 1.$

Thus, the limit evaluates to:

$\lim_{x \to 0} x \csc x = 1.$

Do you want details or have any questions?

Here are 5 related questions to expand your understanding:

1. What is the limit of $\lim_{x \to 0} \frac{\sin x}{x}$?
2. How do you evaluate $\lim_{x \to 0} \frac{1 - \cos x}{x^2}$?
3. What is the Taylor series expansion of $\sin x$ near $x = 0$, and how can it be used to evaluate limits?
4. How can L'Hôpital's Rule be used to evaluate the limit $\lim_{x \to 0} \frac{x}{\sin x}$?
5. What is the limit of $\lim_{x \to 0} \frac{x}{\tan x}$?

Tip: For small angles, $\sin x \approx x$ can be used to approximate trigonometric expressions and evaluate limits.