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 $\infty \times \infty$

$\operatorname{cosec} x-\cot x=(1-\cos x) / \sin x$

$\lim _{x \rightarrow 0} \frac{\operatorname{cosec} x-\cot x}{x}=\lim _{x \rightarrow 0} \frac{1-\cos x}{x \sin x}=\lim _{x \rightarrow 0} \frac{\frac{1-\cos x}{x^{2}}}{\frac{x \sin x}{x^{2}}}=\lim _{x \rightarrow 0} \frac{\frac{1-\cos x}{x^{2}}}{\frac{\sin x}{x}}$

Formula used: $\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}=1 / 2$ and $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$

$\lim _{x \rightarrow 0} \frac{\operatorname{cosec} x-\cot x}{x}=\lim _{x \rightarrow 0} \frac{\frac{1-\cos x}{x^{2}}}{\frac{\sin x}{x}}=\frac{1}{2}$

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