Question

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

Answer 1

${lim}_{x\to \frac{\pi }{4}}\frac{\sin x-\cos x}{x-\pi }={lim}_{x\to \frac{\pi }{4}}\frac{\cos x+\sin x}{1}$

Applying L hospital's rule

$={lim}_{x\to \frac{x}{4}}\sin x+\cos x=\sin \frac{\pi }{4}+\cos \frac{\pi }{4}=\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}=\frac{2}{\sqrt{2}}=\sqrt{2}$