Evaluate _ x ( -x) ( -x)

Question

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

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Answer

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

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