Question

$\underset{x\to 0}{lim}\frac{x(1-\sqrt{1-x^2})}{\sqrt{1-x^2}\left({\sin }^{-1}{\left(x\right)}^3\right)}=$

Answer 1

$\underset{x\to 0}{lim}\frac{x(1-\sqrt{1-x^2})}{{\sin }^{-1}(x{)}^3\sqrt{1-x^2}}$

$\Rightarrow \underset{x\to 0}{lim}\frac{x(1-x^2{)}^{-1/2}(1-\sqrt{1-x^2})}{{\sin }^{-1}(x{)}^3}$

$\Rightarrow \left[\underset{x\to 0}{lim}(1-x^2{)}^{-1/2}\right]\left[\underset{x\to 0}{lim}\frac{x(1-\sqrt{1-x^2})}{{\sin }^{-1}(x{)}^3}\right]$

$\Rightarrow \left[\frac{1}{\sqrt{1-0^2}}\right]\left[\underset{x\to 0}{lim}{x}^3\frac{\frac{1-\sqrt{1-x^2}}{x^2}}{{\sin }^{-1}(x{)}^3}\right]$

$\Rightarrow \left(1\right)\left[\underset{x\to 0}{lim}\frac{x^3}{{\sin }^{-1}(x{)}^3}\right]\left[\underset{x\to 0}{lim}\frac{1-\sqrt{1-x^2}}{x^2}\right]$

$\Rightarrow \left[\underset{x\to 0}{lim}\frac{x^3}{{\sin }^{-1}(x{)}^3}\right]\left[\underset{x\to 0}{lim}\frac{(1-\sqrt{1-x^2})(1+\sqrt{1-x^2})}{x^2(1+\sqrt{1-x^2})}\right]$

$\Rightarrow \left[\underset{x\to 0}{lim}\frac{x^3}{{\sin }^{-1}(x{)}^3}\right]\left[\underset{x\to 0}{lim}\frac{1}{\sqrt{1-x^2}+1}\right]$

$\Rightarrow \left[\underset{x\to 0}{lim}\frac{x^3}{{\sin }^{-1}(x{)}^3}\right]\left[\frac{1}{2}\right]$

$\Rightarrow {\left[\underset{x\to 0}{lim}\frac{x}{{\sin }^{-1}x}\right]}^3\left[\frac{1}{2}\right]$

$\Rightarrow {\left[\underset{x\to 0}{lim}\frac{\sqrt{1-x^2}}{1}\right]}^3\left[\frac{1}{2}\right]$

$\Rightarrow [\sqrt{1-0^2}{]}^3\left[\frac{1}{2}\right]$

$\Rightarrow \frac{1}{2}$.