Question

Solve:
$\underset{x\to 0}{lim}(\csc x-\cot x)$

Answer 1

Given that,

$\underset{x\to 0}{lim}\left(\cos ecx-\cot x\right)$

We know that,

$\begin{array}{c}\cos ecx=\frac{1}{\sin x}\\ \cot x=\frac{\cos x}{\sin x}\end{array}$

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

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

Now applying L'Hospital's rule we get,

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

$=0$.