Question

Evaluate $\underset{x\to \frac{\pi }{4}}{lim}\frac{(sinx-cosx)}{(x-\frac{\pi }{4})}$

Answer 1

put $(x-\frac{\pi }{4})=y$ so that when $x\to \frac{\pi }{4}theny\to 0.$

$\therefore \underset{x\to \frac{\pi }{4}}{lim}\frac{(sinx-cosx)}{(x-\frac{\pi }{4})}$

$=\underset{y\to 0}{lim}\frac{\{sin(\frac{\pi }{4}+y)-cos(\frac{\pi }{4}+y)\}}{y}\left[putting\right(x-\frac{\pi }{4})=y]$

$=\underset{y\to 0}{lim}\frac{\{(\frac{\pi }{4}cosy+cos\frac{\pi }{4}siny)-(cos\frac{\pi }{4}cosy-sin\frac{\pi }{4}siny)\}}{y}$

$=\frac{2}{\sqrt{2}}\times \underset{y\to 0}{lim}\left(\frac{siny}{y}\right)=(\sqrt{2}\times 1)=\sqrt{2}.$