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Evaluate the following limits:

Question:

Evaluate the following limits:

$\lim _{x \rightarrow 0} \frac{\sin (\pi-x)}{\pi(\pi-x)}$

 

Solution:

$=\lim _{x \rightarrow 0} \frac{\sin (\pi-x)}{\pi(\pi-x)}$

$=\lim _{x \rightarrow 0} \frac{\sin x}{\pi(\pi-x)} \times \frac{x}{x}$

$=\lim _{x \rightarrow 0} \frac{\sin x}{x} \times \lim _{x \rightarrow 0} \frac{\pi-(\pi-x)}{\pi(\pi-x)}$

$=1 \times \lim _{x \rightarrow 0}\left(\frac{1}{\pi-x}-\frac{1}{\pi}\right)$

$=\frac{1}{\pi}-\frac{1}{\pi}$

$=0$

$\therefore \lim _{x \rightarrow 0} \frac{\sin (\pi-x)}{\pi(\pi-x)}=0$

 

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