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.

Learn how to evaluate the limit limx->0 x * csc(x) using trigonometric identities and standard limit results. This problem explores key mathematical concepts in limits and trigonometric functions, including the formula limx->0 x/sin(x) = 1. Suitable for students in Grades 11-12.

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