Question

Evaluate the limit:
$\underset{x\to 0}{lim}\frac{\text{cosec}x-\cot x}{x}$

Answer 1

We have,

$\underset{x\to 0}{lim}\frac{\text{cosec}x-\cot x}{x}$

$=\underset{x\to 0}{lim}\frac{\frac{1}{\sin x}-\frac{\cos x}{\sin x}}{x}$

$=\underset{x\to 0}{lim}\frac{1-\cos x}{x\sin x}$

$=\underset{x\to 0}{lim}\frac{2{\sin }^2\frac{x}{2}}{x\sin x}$
$[\because 1-\cos x=2{\sin }^2\frac{x}{2}]$

Dividing numerator and denominator by $x^2,$ we get

$\underset{x\to 0}{lim}\frac{2\times \frac{1}{2}\times \frac{\sin \frac{x}{2}}{\frac{x}{2}}\times \frac{1}{2}\times \frac{\sin \frac{x}{2}}{\frac{x}{2}}}{\frac{\sin x}{x}}$

$=\frac{2\times \frac{1}{2}\times 1\times \frac{1}{2}\times 1}{1}$
$[\because \underset{x\to 0}{lim}\frac{\sin x}{x}=1]$

$=\frac{1}{2}$

$\therefore \underset{x\to 0}{lim}\frac{\text{cosec}x-\cot x}{x}=\frac{1}{2}$