Question

Evaluate $\underset{x\to 0}{lim}\frac{{\cos }^{-1}(1-x)}{\sqrt{x}}$

Answer 1

${lim}_{x\to 0}\frac{{\cos }^{-1}(1-x)}{\sqrt{x}}$

Applying L-Hospital Rule, we get

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

${lim}_{x\to 0}\frac{2\sqrt{x}}{\sqrt{1-1-x^2+2x}}$

${lim}_{x\to 0}\frac{2\sqrt{x}}{\sqrt{2x-x^2}}$

By applying L-Hospital Rule,

${lim}_{x\to 0}\frac{\frac{1}{2\sqrt{x}}\times 2}{\frac{1}{\sqrt{2x-x^2}(2-2x)}}=\frac{\sqrt{2x-x^2}}{\sqrt{x}(1-x)}$

${lim}_{x\to 0}\frac{\sqrt{x}(\sqrt{2}-x)}{\sqrt{x}(1-x)}=\frac{\sqrt{2}}{1}=\sqrt{2}$