Question

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

Answer 1

Now,

$\underset{x\to \pi /4}{lim}\frac{\sqrt{\cos x}-\sqrt{\sin x}}{x-\pi /4}$ $\left(0/0\right)form.$

Now applying L'Hospital's rule we get,

$=\underset{x\to \pi /4}{lim}\frac{\frac{-\sin x}{2\sqrt{\cos x}}-\frac{\cos x}{2\sqrt{\sin x}}}{1}$

$=-\frac{1}{2}(\frac{\sqrt{\sqrt{2}}}{\sqrt{2}}+\frac{\sqrt{\sqrt{2}}}{\sqrt{2}})$

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