Evaluate _ x ( -x) ( -x) - Vidyadip

Question

Evaluate $\lim _{x \rightarrow \pi} \frac{\sin (\pi-x)}{\pi(\pi-x)}$

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Answer

$\text { Let } y=\lim _{x \rightarrow \pi} \frac{\sin (\pi-x)}{\pi(\pi-x)}\left[\frac{0}{0} \text { from }\right]$
$\text { Put } x=\pi+y, \text { as } x \rightarrow \pi, y \rightarrow 0$
$\therefore y=\lim _{y \rightarrow 0} \frac{\sin [\pi-\pi-y]}{\pi[\pi-\pi-y]}=\lim _{y \rightarrow 0} \frac{\sin (-y)}{-\pi y}$
$=\lim _{y \rightarrow 0} \frac{-\sin y}{-\pi y}$
$=\frac{1}{\pi} \lim _{y \rightarrow 0} \frac{\sin y}{y}$
$=\frac{1}{\pi} \times 1$
$=\frac{1}{\pi}$

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