Question

Solve:
$\underset{x\to 0}{lim}(cosecx-cotx)$

Answer 1

We have,

${lim}_{x\to 0}(\csc x-\cot x)$

${lim}_{x\to 0}(\frac{1}{\sin x}-\frac{\cos x}{\sin x})$

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

This is the $\frac{0}{0}$ form.

So, apply L-hospital rule

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

${lim}_{x\to 0}\left(\frac{\sin x}{\cos x}\right)$

${lim}_{x\to 0}\left(\tan x\right)$

$=\tan 0=0$

Hence, this is the answer.