Given,
\(\lim\limits_{x\to 0} \frac{cosec x - cot x}x\)
\(= \lim\limits_{x\to 0} \cfrac{(\frac 1{\sin x})- \frac{\cos x}{\sin x}}x\)
\(= \lim\limits_{x\to 0} \frac{1 - \cos x}{x . \sin x}\)
\(= \lim\limits_{x\to 0} \cfrac{2\frac{\sin^2x}x}{x.2\frac{\sin x}2 \frac{\cos x}2}\)
\(= \lim\limits_{x\to 0} \cfrac{\frac {\tan x}2}x\)
\(= \lim\limits_{x\to 0}\cfrac{\frac{\tan x}2}{\frac x2}.\frac 12\)
\(= \frac 12\) \(\left[\because \lim\limits_{\theta\to 0} \frac{\tan \theta}\theta = 1\right]\)
