Question

limit X tends to pi by 4 sin x minus cos x by x minus 5 by 4 equals to

Answer 1

Solution:

Result used:

$\lim_{\theta\to\:0}\frac{sin\theta}{\theta}=1$

$\lim_{x\to\frac{\pi}{4}}\frac{sinx-cosx}{x-\frac{\pi}{4}}$

$=\lim_{x\to\frac{\pi}{4}}\sqrt2[\frac{sinx\frac{1}{\sqrt2}-cosx\frac{1}{\sqrt2}}{x-\frac{\pi}{4}}]$

$=\lim_{x\to\frac{\pi}{4}}\sqrt2[\frac{sinx.cos\frac{\pi}{4}-cosx.sin\frac{\pi}{4}}{x-\frac{\pi}{4}}]$

$=\lim_{x\to\frac{\pi}{4}}\sqrt2[\frac{sin(x-\frac{\pi}{4})}{x-\frac{\pi}{4}}]$

take $t=x-\frac{\pi}{4}$

as ${x\to\frac{\pi}{4}}, {t\to\:0}$

$=\sqrt{2}\lim_{t\to\:0}\frac{sint}{t}\\=\sqrt{2}.1\\=\sqrt{2}$