$\displaystyle \mathop {\lim }\limits_{x \to \frac{\pi }{2}} \frac{{\cot x - \cos x}}{{{{\left( {\frac{\pi }{2} - x} \right)}^3}}}$

Ex$\begin{array}{l}\left.\lim _{n \rightarrow \frac{\pi}{2}} \frac{\cot n-\cos n}{\left(\frac{\pi}{2}-n\right)^{3}} \text { [० আকার }\right] \\ =\lim _{x \rightarrow \frac{\pi}{2}} \frac{-\operatorname{cosec}^{2} x+\sin m}{3\left(\frac{\pi}{2}-n\right)^{2}(-1)} \text { [L'Ho, pital] } \\ =\lim _{n \rightarrow \frac{\pi}{2}} \frac{-2 \operatorname{cosec} n(\cot x \operatorname{cosec} n)+\cos n}{6\left(\frac{\pi}{2}-n\right)} \\\end{array}$

$\begin{array}{l}=\lim _{m \rightarrow \frac{\pi}{2}} \frac{2 \operatorname{cosec}^{2} x \cot x+\cos n}{6\left(\frac{\pi}{2}-m\right)} \\ =\lim _{n \rightarrow \frac{\pi}{2}} \frac{2 \frac{1}{\cos ^{2} n} \times \frac{\cos n}{\sin n}+\cos n}{6\left(\frac{\pi}{2}-m\right)} \\ =\lim _{x \rightarrow \frac{\pi}{2}} \frac{4 \operatorname{cosec} 2 x+\cos x}{6\left(\frac{\pi}{2}-x\right)} \\\end{array}$

$\begin{array}{l}=\lim _{x \rightarrow \frac{\pi}{2}} \frac{4(-\cot 2 x \operatorname{cosec} 2 x) 2-\sin x}{-6} \\ =\frac{8(-\cot \pi \operatorname{cosec} \pi)-\sin \pi / 2}{-6} \\ =\frac{-1}{-6}=\frac{1}{6} \text { Ans }\end{array}$