Question

Evaluate : ${lim}_{x\to \frac{\pi }{4}}\frac{\sqrt{\cos x}-\sqrt{\sin x}}{x-\frac{\pi }{4}}$

Answer 1

$\underset{x\to \frac{\pi }{4}}{lim}\frac{\sqrt{\cos x}-\sqrt{\sin x}}{x-\frac{\pi }{4}}$

$=\underset{x\to \frac{\pi }{4}}{lim}\left(\frac{\frac{-\sin x}{2\sqrt{\cos x}}-\frac{\cos x}{2\sqrt{\sin x}}}{1}\right)$

$=\underset{x\to \frac{\pi }{4}}{lim}(\frac{-\sin x}{2\sqrt{\cos x}}-\frac{\cos x}{2\sqrt{\sin x}})$

$=(\frac{-\sin \frac{\pi }{4}}{2\sqrt{\cos \frac{\pi }{4}}}-\frac{\cos \frac{\pi }{4}}{2\sqrt{\sin \frac{\pi }{4}}})$

$=-\frac{1}{\sqrt[4]{42}}$