Question

$Li{m}_{x\to 0}\frac{x(1-\sqrt{1-x^2})}{\sqrt{1-x^2}\left(si{n}^{-1}\right(x{)}^3)}=$

Answer 1

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

Rationalising , we get

${lim}_{x\to 0}\frac{x(1-\sqrt{1-x^2})\times (1+\sqrt{1-x^2})}{\sqrt{1-x^2}(1+\sqrt{1+x^2}){\sin }^{-1}\left(x^3\right)}={lim}_{x\to 0}\frac{x(1-1+x^2)}{\sqrt{1-x^2}(1+\sqrt{1+x^2}){\sin }^{-1}\left(x^3\right)}={lim}_{x\to 0}\frac{1}{\sqrt{1-x^2}(1+\sqrt{1-x^2})\times \frac{{\sin }^{-1}\left(x^3\right)}{x^3}}=\frac{1}{1\times (1+1)\times 1}=\frac{1}{2}\therefore L=\frac{1}{2}$