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

Question

Evaluate $\mathop {\lim }\limits_{x \to \pi } \frac{{\sin (\pi - x)}}{{\pi (\pi - x)}}$

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Answer

Let y= $\mathop {\lim }\limits_{x \to \pi } \frac{{\sin (\pi - x)}}{{\pi (\pi - x)}}\left[ {\frac{0}{0}{\text{from}}} \right]$
Put $x = \pi + y$, as $x \to \pi ,\;y \to 0$
$\therefore y=\;\mathop {\lim }\limits_{y \to 0} \frac{{\sin [\pi - \pi - y]}}{{\pi [\pi - \pi - y]}}=\mathop {\lim }\limits_{y \to 0} \frac{{\sin ( - y)}}{{ - \pi y}}$
$ = \mathop {\lim }\limits_{y \to 0} \frac{{ - \sin y}}{{ - \pi y}} = \frac{1}{\pi }\mathop {\lim }\limits_{y \to 0} \frac{{\sin y}}{y} = \frac{1}{\pi } \times1 = \frac{1}{\pi }$

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