Question

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

Answer 1

$$\begin{gathered} \underset{x\to 0}{\lim }\left[\frac{\mathrm{cosec}x-\cot x}{x}\right] \\ =\underset{x\to 0}{\lim }\left[\frac{{\displaystyle \frac{1}{\sin x}}-{\displaystyle \frac{\cos x}{\sin x}}}{x}\right] \\ =\underset{x\to 0}{\lim }\left[\frac{1-\cos x}{x\sin x}\right] \\ =\underset{x\to 0}{\lim }\left[\frac{2{\sin }^2\left({\displaystyle \frac{x}{2}}\right)}{x\sin x}\right] \\ =\underset{x\to 0}{\lim }\left[\frac{2{\sin }^2\left({\displaystyle \frac{x}{2}}\right)}{{\displaystyle {\left(\frac{x}{2}\right)}^2}}\times \frac{{\displaystyle \frac{x^2}{4}}}{{\displaystyle \frac{x\sin x}{x\times x}}\times x^2}\right] \\ =2\times \frac{1}{4} \\ =\frac{1}{2} \end{gathered}$$