Question

$\underset{x\to \pi /2}{lim}\frac{(1-\tan x/2)(1-\sin x)}{(1-\tan x/2){(\pi -2x)}^3}$ is equal to

• A. $\infty$
• B. $1/8$
• C. $0$
• D. $1/32$

Answer 1

Answer: (D)

$1/32$

We have
${lim}_{x\to \pi /2}\left(\frac{1-\tan x/2}{1+\tan x/2}\right)\left\{\frac{1-\sin x}{{(\pi -2x)}^3}\right\}$
$=\frac{1}{64}\underset{x\to \pi /2}{lim}\frac{\tan (\frac{\pi }{4}-\frac{x}{2})}{(\frac{\pi }{4}-\frac{x}{2})}\frac{[1-\cos (\pi /2-x)]}{{(\frac{\pi }{4}-\frac{x}{2})}^2}$
$=1/64\times 1\times 2=1/32$