Evaluate the following limits:

To Find: Limits

NOTE: First Check the form of imit. Used this method if the limit is satisfying any one from 7 indeterminate form.

In this Case, indeterminate Form is $0 \times \infty$

Formula used: $\lim _{x \rightarrow 0} \frac{x}{\tan x}=1$

So, by using the above formula, we have

$\lim _{x \rightarrow 0} x \cot 2 x=\lim _{x \rightarrow 0} \frac{2 x}{2 \tan 2 x}=\frac{1}{2}$

Therefore, $\lim _{x \rightarrow 0} x \cot x=\frac{1}{2}$