Question

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

Answer 1

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

Apply L-Hospital rule.

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

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