Question

Evalaute

$\underset{x\to \frac{\pi }{4}}{lim}\frac{cosx-sinx}{\frac{\pi }{4}-x}=$

Answer 1

$\underset{x\to \frac{x}{4}}{lim}\frac{\cos x-\sin x}{\frac{\pi }{4}-x}$ which is an indeterminate from $\frac{0}{0}$

using L Hospital rule, $f\left(x\right)=\cos x-\sin x$

$f\left(x\right)=-\sin x-\cos x$

$g\left(x\right)=\frac{\pi }{4}-x,g^{\prime }\left(x\right)=-1$

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

$=\sin \frac{\pi }{4}+\cos \frac{\pi }{4}$

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

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

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

$=\sqrt{2}$