Question:
Evaluate the Given limit: \(\lim_{x\rightarrow \pi}\) \(\frac{sin(\pi-x)}{\pi(\pi-x)}\)
Evaluate the Given limit: \(\lim_{x\rightarrow \pi}\) \(\frac{sin(\pi-x)}{\pi(\pi-x)}\)
Updated On: Oct 23, 2023
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Solution and Explanation
\(\lim_{x\rightarrow \pi}\) \(\frac{sin(\pi-x)}{\pi(\pi-x)}\)
It is seen that \(x\rightarrow \pi\)\(\Rightarrow\) (\(\pi\) - x ) \(\rightarrow\)0
∴\(\lim_{x\rightarrow \pi}\) \(\frac{sin(\pi-x)}{\pi(\pi-x)}\) = \(\frac{1}{\pi}\) \(\lim_{\pi-x\rightarrow \pi}\) \(\frac{sin(\pi-x)}{(\pi-x)}\)
= \(\frac{1}{\pi}\)\(\times\)1 [\(\lim_{y\rightarrow \pi}\)\(\frac{sin\,y}{y}\) = 1]
=\(\frac{1}{\pi}\)
It is seen that \(x\rightarrow \pi\)\(\Rightarrow\) (\(\pi\) - x ) \(\rightarrow\)0
∴\(\lim_{x\rightarrow \pi}\) \(\frac{sin(\pi-x)}{\pi(\pi-x)}\) = \(\frac{1}{\pi}\) \(\lim_{\pi-x\rightarrow \pi}\) \(\frac{sin(\pi-x)}{(\pi-x)}\)
= \(\frac{1}{\pi}\)\(\times\)1 [\(\lim_{y\rightarrow \pi}\)\(\frac{sin\,y}{y}\) = 1]
=\(\frac{1}{\pi}\)
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