Question

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

Answer 1

Here $\underset{x\to 0}{\text{lim}}(cosecx-cotx)$

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

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

= $\underset{x\to 0}{\text{lim}}(cosecx-cotx)$

= $\underset{x\to 0}{\text{lim}}\text{tan}\frac{x}{2}$ = 0.