Question

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

Answer 1

$\underset{x\to \frac{\pi }{4}}{lim}\frac{\cos x-\sin x}{\frac{\pi }{4}-x}=\frac{0}{0}$ form

Applying L-Hospital rule

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

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

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