Question

Evaluate $\underset{x\to \frac{\pi }{4}}{lim}\frac{\cos x-\sin x}{(\frac{\pi }{4}-x)(\cos x+\sin x)}$

Answer 1

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

Using L' Hospital's Rule, we get

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

$=\frac{(-1/\sqrt{2}-1/\sqrt{2})}{(-1)(1/\sqrt{2}+1/\sqrt{2})+0}$

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

$=1$

$\therefore \underset{x\to \pi /4}{lim}\frac{\cos x-\sin x}{(\pi /4-x)(\cos x+\sin x)}=1$.