Question

Evaluate
$\underset{x\to 0}{lim}\frac{\sqrt{2}-\sqrt{1+\cos x}}{{\sin }^{2}x}$

Answer 1

Solution -

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

Apply L - Hospital Rule -

${lim}_{x\to 0}\frac{-1(-sinx)}{2\frac{\sqrt{1+cosx}}{2sinxcosx}}$

${lim}_{x\to 0}\frac{1}{4cosx\sqrt{1+cosx}}$

$=\frac{1}{4\sqrt{2}}$