Question

The value of $\underset{x\to \pi /2}{lim}\left[\frac{x-\frac{\pi }{2}}{\cos x}\right]$ (where , [x] denotes greatest integer function) is

• A. $-1$
• B. $0$
• C. $-2$
• D. Does not exist

Answer 1

The correct option is

A $-1$

${lim}_{x\to \frac{\pi }{2}}\left[\frac{x-\frac{\pi }{2}}{\cos x}\right]$

Put $x=\frac{\pi }{2}$ in ${lim}_{x\to \frac{\pi }{2}}\left[\frac{x-\frac{\pi }{2}}{\cos x}\right]$

$=\left[\frac{\frac{\pi }{2}-\frac{\pi }{2}}{\cos \frac{\pi }{2}}\right]$

$=\frac{0}{0}$

So, By using L hospital rule, differentiate both the numerator and the denominator with respect to $x$

${lim}_{x\to \frac{\pi }{2}}\left[\frac{\frac{d}{dx}(x-\frac{\pi }{2})}{\frac{d}{dx}\cos x}\right]$

$={lim}_{x\to \frac{\pi }{2}}\left[\frac{1-0}{-\sin x}\right]$

$={lim}_{x\to \frac{\pi }{2}}\left[\frac{-1}{\sin x}\right]$

$=\frac{-1}{\sin \frac{\pi }{2}}$

$=-1$