Question

$\underset{x\to \frac{\pi }{2}}{lim}\frac{\left[\frac{x}{2}\right]}{\ln \left(\sin x\right)}$, (where $[.]$ denotes the greatest integer function)

• A. does not exist
• B. equals $1$
• C. equals $0$
• D. equals $-1$

Answer 1

The correct option is A does not exist

$\underset{x\to \frac{\pi }{2}}{lim}\frac{\left[\frac{x}{2}\right]}{ln\sin x}$

Considering left hand limit

$\underset{x\to \frac{\pi }{2}-}{lim}$ $\frac{\left[\frac{x}{2}\right]}{ln\sin x}=\frac{\left[{\frac{\pi }{4}}^{-}\right]}{ln\sin \frac{\pi }{2}}=\infty$

Also, right hand limit $=\infty$

$\therefore$ We say limit does not exists.