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!