Evaluate the Given limit:

At $x=0$, the value of the given function takes the form $\infty-\infty$.

Now,

$\lim _{x \rightarrow 0}(\operatorname{cosec} x-\cot x)$

$=\lim _{x \rightarrow 0}\left(\frac{1}{\sin x}-\frac{\cos x}{\sin x}\right)$

$=\lim _{x \rightarrow 0}\left(\frac{1-\cos x}{\sin x}\right)$

$=\lim _{x \rightarrow 0} \frac{\left(\frac{1-\cos x}{x}\right)}{\left(\frac{\sin x}{x}\right)}$

$=\frac{\lim _{x \rightarrow 0} \frac{1-\cos x}{x}}{\lim _{x \rightarrow 0} \frac{\sin x}{x}}$

$=\frac{0}{1} \quad\left[\lim _{x \rightarrow 0} \frac{1-\cos x}{x}=0\right.$ and $\left.\lim _{x \rightarrow 0} \frac{\sin x}{x}=1\right]$

$=0$