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

  1. (1) $\pi$

  2. (2) 0

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

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


Correct Option: , 4

Solution:

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

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