Evaluate: _ x 3 1- 6 2 ( 3 -x ) - Vidyadip

Question

Evaluate:
$\lim\limits_{\text{x} \rightarrow \frac{\pi}{3}}\frac{\sqrt{1-\cos6}}{\sqrt{2}\bigg(\frac{\pi}{3}-\text{x}\bigg)} $

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Answer

Given that $\lim\limits_{\text{x} \rightarrow \frac{\pi}{3}}\frac{\sqrt{1-\cos6}}{\sqrt{2}\bigg(\frac{\pi}{3}-\text{x}\bigg)} $
$=\lim\limits_{\text{x} \rightarrow\frac{\pi}{3}}\frac{\sqrt{2\sin^{2}3\text{x}}}{\sqrt{2}\bigg(\frac{\pi}{3}-\text{x}\bigg)} $
$=\lim\limits_{\text{x} \rightarrow\frac{\pi}{3}}\frac{\sqrt{2}\sin3\text{x}}{\sqrt{2}\bigg(\frac{\pi-3\text{x}}{3}\bigg)}$
$=\lim\limits_{\text{x} \rightarrow\frac{\pi}{3}}\frac{3.\sin(\pi-3\text{x})}{\pi-3\text{x}} $
$=3\Big[\therefore\lim\limits_{\text{x} \rightarrow0}\frac{\sin\text{x}}{\text{x}}=1\Big]$
Hence, the required answer is 3.

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