We need to evaluate the limit:

\[
\lim_{{x \to 0}} 2x \cdot \cot(x)
\]

We know that \(\cot(x) = \frac{1}{\tan(x)}\), so the expression becomes:

\[
\lim_{{x \to 0}} 2x \cdot \frac{1}{\tan(x)} = \lim_{{x \to 0}} \frac{2x}{\tan(x)}
\]

Using the standard limit identity:

\[
\lim_{{x \to 0}} \frac{x}{\tan(x)} = 1
\]

We then have:

\[
\lim_{{x \to 0}} \frac{2x}{\tan(x)} = 2 \times 1 = 2
\]

Thus, the solution is:

\[
\lim_{{x \to 0}} 2x \cdot \cot(x) = 2
\]

Would you like any further explanation or details on this?

Here are five related questions:
1. What is the limit of \( \lim_{{x \to 0}} \frac{x}{\sin(x)} \)?
2. How do you find the limit of a product of functions as \(x \to 0\)?
3. What is the behavior of \( \cot(x) \) as \( x \to 0 \)?
4. Can L'Hôpital's Rule be applied to find this limit?
5. How does the Taylor series expansion of trigonometric functions help with limits?

**Tip:** Memorizing common trigonometric limits, like \( \lim_{{x \to 0}} \frac{\sin(x)}{x} = 1 \), can greatly speed up solving such problems!

Discover how to find the limit of 2x * cot(x) as x approaches 0. This trigonometric limit problem involves concepts like limits, standard trigonometric identities, and uses the identity lim(x -> 0) (x/tan(x)) = 1. Suitable for students in Grades 10-12.

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