Evaluate: _ X 6 [ 3 x - x x- 6 ] - Vidyadip

Question

Evaluate:$\lim\limits_{X\rightarrow \frac{\pi}{6}} \bigg[ \frac{\sqrt{3}\sin x - \cos x}{x- \frac{\pi}{6}}\bigg]$

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Answer

$\lim\limits_{X\rightarrow \frac{\pi}{6}} \Bigg[ \frac{\sqrt{3}\sin x - \cos x}{x- \frac{\pi}{6}}\Bigg] \lim\limits_{X\rightarrow \frac{\pi}{6}} \Bigg[2\frac{\Big( \frac{\sqrt{3}}{2}.\sin x-\frac{1}{2}\cos x\Big)}{x-\frac{\pi}{6}}\Bigg]$$\lim\limits_{X\rightarrow \frac{\pi}{6}} \Bigg[\frac{2\big(\cos \frac{\pi}{6}.\sin x-\sin\frac{\pi}{6}.\cos x\big)}{x-\frac{\pi}{6}}\Bigg]$
$2\lim\limits_{X\rightarrow\frac{\pi}{6}}\frac{\sin\big(x-\frac{\pi}{6}\big)}{\big(x - \frac{\pi}{6}\big)}$

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