Evaluate the following limits in Exercise: _ x ( -x) ( -x) - Vidyadip

Question

Evaluate the following limits in Exercise:$\lim\limits_{\text{x}\rightarrow\pi}\frac{\sin(\pi-\text{x})}{\pi(\pi-\text{x})}$

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Answer

$\lim\limits_{\text{x}\rightarrow\pi}\frac{\sin(\pi-\text{x})}{\pi(\pi-\text{x})}$ It is seen that $\text{x} →\pi ⇒ (\pi –\text{x}) → 0$$\lim\limits_{\text{x}\rightarrow\pi}\frac{\sin(\pi-\text{x})}{\pi(\pi-\text{x})}=\frac{1}{\pi}\lim\limits_{(\text{x}-\pi)\rightarrow0}\frac{\sin(\pi-\text{x})}{(\pi-\text{x})}$
$=\frac{1}{\pi}\times1$ $\bigg[\lim\limits_{\text{y}\rightarrow0}\frac{\sin(\text{y})}{\text{y}}=1\bigg]$
$=\frac{1}{\pi}$

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