Question

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

Answer 1

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

It becomes $\frac{0}{0}$ form

Multiply and Divide the Numerator by $\sqrt{2}$
$=\underset{x\to \frac{\pi }{4}}{lim}\frac{\sqrt{2}(\frac{1}{\sqrt{2}}\cos x-\frac{1}{\sqrt{2}}\sin x)}{(\frac{\pi }{4}-x)(\cos x+\sin x)}$

$=\underset{x\to \frac{\pi }{4}}{lim}\frac{\sqrt{2}(\sin \frac{\pi }{4}\cos x-\cos \frac{\pi }{4}\sin x)}{(\frac{\pi }{4}-x)(\cos x+\sin x)}$

$=\underset{x\to \frac{\pi }{4}}{lim}\frac{\sqrt{2}\sin (\frac{\pi }{4}-x)}{(\frac{\pi }{4}-x)(\cos x+\sin x)}$
$\because \sin (A-B)=\sin A\cos B-\cos A\sin B$

$=\underset{x\to \frac{\pi }{4}}{lim}\sqrt{2}\times \underset{x\to \frac{\pi }{4}}{lim}\frac{\sin (\frac{\pi }{4}-x)}{(\frac{\pi }{4}-x)}\times \frac{1}{(\cos x+\sin x)}$

$=\sqrt{2}\times 1\times \frac{1}{(\cos \frac{\pi }{4}+\sin \frac{\pi }{4})}$

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

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