Question

Evaluate the Given limit: $\underset{x\to 0}{lim}(\cos ecx-\cot x)$

Answer 1

$\underset{x\to 0}{lim}(\cos ecx-\cot x)$
$=\underset{x\to 0}{lim}(\frac{1}{\sin x}-\frac{\cos x}{\sin x})$
$=\underset{x\to 0}{lim}\left(\frac{1-\cos x}{\sin x}\right)$
$=\underset{x\to 0}{lim}\frac{\left(\frac{1-\cos x}{x}\right)}{\left(\frac{\sin x}{x}\right)}$
$=\frac{{lim}_{x\to 0}\frac{1-\cos x}{x}}{{lim}_{x\to 0}\frac{\sin x}{x}}=\frac{0}{1}=0$
$[\because \underset{x\to 0}{lim}\frac{1-\cos x}{x}=0\text{and}\underset{x\to 0}{lim}\frac{\sin x}{x}=1]$