Mathematics - Limits Question with Solution | TestHub

Given $L=\underset{x\to 0^{+}}{\lim }\frac{{\cos }^{-1}\left(x-[x{]}^2\right)·{\sin }^{-1}\left(x-[x{]}^2\right)}{x-x^3}$

$\because x\in [0,1)\Rightarrow \left[x\right]=0$

Then $L=\underset{x\to 0^{+}}{\lim }\frac{{\cos }^{-1}x}{\left(1-x^2\right)}\times \underset{x\to 0^{+}}{\lim }\frac{{\sin }^{-1}x}{x}$

$L=\frac{\mathrm{\pi }}{2}\times \underset{x\to 0^{+}}{\lim }\frac{1}{\sqrt{1-x^2}} \left(\text{Apply L'Hospital rule for }\frac{0}{0}\text{ form}\right)$

$L=\frac{\mathrm{\pi }}{2}$