Question

Evaluate the Given limit: $\lim_{x\rightarrow \pi}$ $\frac{sin(\pi-x)}{\pi(\pi-x)}$

Answer 1

$\lim_{x\rightarrow \pi}$ $\frac{sin(\pi-x)}{\pi(\pi-x)}$
It is seen that x\rightarrow \pi$$\Rightarrow ($\pi$ - x ) $\rightarrow$0
∴$\lim_{x\rightarrow \pi}$ $\frac{sin(\pi-x)}{\pi(\pi-x)}$ = $\frac{1}{\pi}$ $\lim_{\pi-x\rightarrow \pi}$ $\frac{sin(\pi-x)}{(\pi-x)}$
= \frac{1}{\pi}$$\times1 [\lim_{y\rightarrow \pi}$$\frac{sin\,y}{y} = 1]
=$\frac{1}{\pi}$