Question

$\underset{x\to \frac{\pi }{2}}{lim}\left[\frac{x-\frac{\pi }{2}}{\cos x}\right]$ is equal to (where $[.]$ respresent greatest integer function)

• A. $-1$
• B. $0$
• C. $-2$
• D. $doesnotexits$

Answer 1

The correct option is A $-1$

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

$\underset{x\to \frac{\pi }{2}}{lim}\left[\frac{x-\frac{\pi }{2}}{\cos x}\right]=\frac{0}{0}$

It is the indeterminate form, so applying L’Hospitals rule,

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

$=\left[\frac{1}{-1}\right]$

$=-1$