Evaluate: _ x 6 3 - x- 6 - Vidyadip

Question

Evaluate:
$\lim\limits_{\text{x} \rightarrow\frac{\pi}{6}}\frac{\sqrt{3}\sin\text{x}-\cos\text{x}}{\text{x}-\frac{\pi}{6}}$

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Answer

Given that $\lim\limits_{\text{x} \rightarrow\frac{\pi}{6}}\frac{\sqrt{3}\sin\text{x}-\cos\text{x}}{\text{x}-\frac{\pi}{6}}$
$=\lim\limits_{\text{x} \rightarrow\frac{\pi}{6}}\frac{2\Big[\frac{\sqrt{3}}{2}\sin\text{x}-\frac{1}{2}\cos\text{x}\Big]}{\text{x}-\frac{\pi}{6}}$
$=\lim\limits_{\text{x} \rightarrow\frac{\pi}{6}}\frac{2\big[\cos\frac{\pi}{6}\sin\text{x}-\sin\frac{\pi}{6}\cos\text{x}\big]}{\text{x}-\frac{\pi}{6}}$
$=\lim\limits_{\text{x} \rightarrow\frac{\pi}{6}}\frac{2\sin\big(\text{x}-\frac{\pi}{6}\big)}{\big(\text{x}-\frac{\pi}{6}\big)}$ $\bigg[ \therefore\lim\limits_{\text{x} \rightarrow\frac{\pi}{6}}\frac{\sin\text{x}}{\text{x}}=1\bigg] $
$=2.1=2$
Hence, the required answer is 2.

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