Question

Evaluate the following limit:
$\underset{x\to \pi }{\text{lim}}\frac{\text{sin}(\pi -x)}{\pi (\pi -x)}$

Answer 1

Here $\underset{x\to \pi }{\text{lim}}\frac{\text{sin}(\pi -x)}{\pi (\pi -x)}\left[\frac{0}{0}\text{Form}\right]$

Put x = $\pi$ + y, as x $\to \pi ,y\to 0$

$\therefore \underset{y\to 0}{\text{lim}}\frac{\text{sin}[\pi -\pi -y]}{\pi [\pi -\pi -y]}=\underset{y\to 0}{\text{lim}}\frac{\text{sin (-y)}}{-\pi y}$

=$\underset{y\to 0}{\text{lim}}\frac{\text{- sin y}}{-\pi y}=\frac{1}{\pi }\underset{y\to 0}{\text{lim}}\frac{\text{sin y}}{y}=\frac{1}{\pi }\times 1=\frac{1}{\pi }$.