To evaluate the limit of \( x \cot(x) \) as \( x \to 0 \), we proceed as follows:

We know that:
\[
\cot(x) = \frac{\cos(x)}{\sin(x)}
\]

Thus, the expression \( x \cot(x) \) becomes:
\[
x \cot(x) = x \cdot \frac{\cos(x)}{\sin(x)} = \frac{x \cos(x)}{\sin(x)}
\]

Now, we evaluate the limit:
\[
\lim_{x \to 0} \frac{x \cos(x)}{\sin(x)}
\]

We can split this into two parts:
\[
\lim_{x \to 0} \frac{x}{\sin(x)} \cdot \lim_{x \to 0} \cos(x)
\]

From the standard limit, we know:
\[
\lim_{x \to 0} \frac{x}{\sin(x)} = 1
\]
and
\[
\lim_{x \to 0} \cos(x) = 1
\]

Therefore:
\[
\lim_{x \to 0} x \cot(x) = 1 \cdot 1 = 1
\]

So, the value of the limit is:
\[
\boxed{1}
\]

Would you like more details or have any questions?

### Related questions:
1. How do you evaluate limits involving trigonometric functions in general?
2. What is L'Hopital's Rule, and can it be used for this limit?
3. How do small-angle approximations help in limit evaluations?
4. What is the limit \( \lim_{x \to 0} \frac{\sin(x)}{x} \), and why is it important?
5. How does the continuity of cosine and sine functions affect their limits?

**Tip:** Remember that \( \lim_{x \to 0} \frac{\sin(x)}{x} = 1 \) is a crucial result for evaluating many trigonometric limits.

Learn how to evaluate the limit of x cot(x) as x approaches 0. This problem explores important concepts in limits and trigonometry, such as the standard trigonometric limits and the continuity of functions. Suitable for students in Grades 10-12.

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