#### Find the value of $\mathrm{ \lim _{x \rightarrow \pi / 2} \frac{\cot (x)-\cos (x)}{(\pi-2 x)^3} }$  Option: 1 Option: 2 Option: 3 Option: 4

$\mathrm{ =\lim _{x \rightarrow \pi / 2} \frac{\cos (x)(1-\sin (x))}{\sin (x)(\pi-2 x)^3} }$

$\mathrm{ =\lim _{x \rightarrow \pi / 2} \frac{\cos (x)(1-\sin (x))}{x \frac{\sin (x)}{x}(\pi-2 x)^3} }$

$\mathrm{ =\lim _{x \rightarrow \pi / 2} \frac{\cos (x)(1-\sin (x))}{x(\pi-2 x)^3} }$

Now; let $\mathrm{ x-\pi / 2=h }$. Then when $\mathrm{ x \rightarrow \pi / 2, h \rightarrow 0 }$

$\mathrm{ =\lim _{h \rightarrow 0} \frac{\cos (\pi / 2+h)(1-\sin (\pi / 2+h))}{(h+\pi / 2)(\pi-2(\pi / 2+h))^3} }$

$\mathrm{ =\lim _{h \rightarrow 0} \frac{-\sin (h)(1-\cos (h))}{(h+\pi / 2)(-2 h)^3} }$

$\mathrm{ =\lim _{h \rightarrow 0} \frac{1}{8} \frac{\sin (h)}{h} \frac{(1-\cos (h))}{\left(h^2\right)(h+\pi / 2)} }$

$\mathrm{ =\lim _{h \rightarrow 0} \frac{1}{8} \frac{(1-\cos (h))}{\left(h^2\right)(h+\pi / 2)} }$

$\mathrm{ =\lim _{h \rightarrow 0} \frac{1}{8} \frac{\left(2 \sin ^2(h / 2)\right)}{\left(h^2\right)(h+\pi / 2)} }$

$\mathrm{ =\lim _{h \rightarrow 0} \frac{1}{4}\left(\frac{\sin (h / 2)}{(h / 2)}\right)^2 \frac{1}{4(h+\pi / 2)} }$

$\mathrm{ =\lim _{h \rightarrow 0} \frac{1}{16} \frac{1}{(h+\pi / 2)} }$

Hence option 1 is correct.