Question

The value of $\underset{x\to \pi /2}{lim}(x\tan x-\frac{\pi }{2}\sec x)$ equals to

• A. $-1$
• B. $0$
• C. $1$
• D. none of these

Answer 1

The correct option is A $-1$

Let, $L=\underset{x\to \pi /2}{lim}(x\tan x-\frac{\pi }{2}\sec x)$

$=\underset{x\to \pi /2}{lim}\left(\frac{2x\sin x-\pi }{2\cos x}\right),\frac{0}{0}$ form

Thus using L-Hospital rule,

$L=\underset{x\to \pi /2}{lim}\left(\frac{2\sin x+2x\cos x-0}{-2\sin x}\right)=\frac{2\left(0\right)+0}{-2\left(0\right)}=-1$