The value of
Question:

The value of $\lim _{x \rightarrow 0^{+}} \frac{\cos ^{-1}\left(x-[x]^{2}\right) \cdot \sin ^{-1}\left(x-[x]^{2}\right)}{x-x^{3}}$, where

$[\mathrm{x}]$ denotes the greatest integer $\leq \mathrm{x}$ is :

1. $\pi$

2. 0

3. $\frac{\pi}{4}$

4. $\frac{\pi}{2}$

Correct Option: , 4

Solution:

$\lim _{x \rightarrow 0^{+}} \frac{\cos ^{-1} x}{\left(1-x^{2}\right)} \times \frac{\sin ^{-1} x}{x}=\frac{\pi}{2}$