Question

$\displaystyle\lim_{x \to \frac{\pi}{2}} \frac{\cot x - \cos x}{\left(\pi - 2x\right)^{3}} $ equals :

• A. $\frac{1}{16}$
• B. $\frac{1}{8}$
• C. $\frac{1}{4}$
• D. $\frac{1}{24}$

Answer 1

Answer: (A)

$\displaystyle\lim_{x \to \frac{\pi}{2}} \frac{\cot x - \cos x}{\left(\pi - 2x\right)^{3}} $
Put, $\frac{\pi}{2}-x = t$
$\displaystyle\lim_{t \to 0}\frac{\tan \,t-\sin \,t}{8t^{3}}$
$=\displaystyle\lim_{t \to 0}\frac{\sin t\cdot 2 \sin^{2} \frac{t}{2}}{8t^{3}}$
$ = \frac{1}{16}.$