Question

$\underset{x\to -1}{lim}$ $\frac{\sqrt{\pi }-\sqrt{arccosx}}{\sqrt{x+1}}$

Answer 1

$\underset{x\to -1}{lim}\frac{\sqrt{\pi }-\sqrt{\arccos x}}{\sqrt{x+1}}$

Applying L'hospital rule

$\underset{x\to -1}{lim}\frac{\frac{1}{2\sqrt{\arccos x}\sqrt{1-x^2}}}{\frac{1}{2\sqrt{x+1}}}=\underset{x\to -1}{lim}\frac{\sqrt{1-x}}{\sqrt{\arccos x}}=\sqrt{\frac{2}{\pi }}$