Question

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

Answer 1

$\underset{x\to \pi /4}{lim}\frac{\sin x-\cos x}{x-\pi /4}=$undefined form

Applying L'Hospital Rule

$\underset{x\to \pi /4}{lim}\frac{\cos x+\sin x}{1-0}=\frac{\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}}{1}=\frac{2}{\sqrt{2}}=\sqrt{2}$