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 $\frac{0}{0}$

By using $L$ hospital Rule,

Differtiate both sides w.r.t $x$

So $\lim _{x \rightarrow \frac{\pi}{4}} \frac{\operatorname{cosec}^{2} x-2}{\cot x-1}=\lim _{x \rightarrow \frac{\pi}{4}} \frac{2 \operatorname{cosec} x(-\operatorname{cosec} x \cot x)-0}{-\operatorname{cosec}^{2} x-0}=\lim _{x \rightarrow \frac{\pi}{4}} \frac{2 \operatorname{cosec} x(-\operatorname{cosec} x \cot x)}{-\operatorname{cosec}^{2} x}=\lim _{x \rightarrow \frac{\pi}{4}} 2 \cot x=2$

Therefore, $\lim _{x \rightarrow \frac{\pi}{4}} \frac{\operatorname{cosec}^{2} x-2}{\cot x-1}=2$