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Question: \[\lim_{x \rightarrow \pi/6}\left\lbrack \frac{3\sin x - \sqrt{3}\cos x}{6x - \pi} \right\rbrack\]...

lim⁡x→π/6[3sin⁡x−3cos⁡x6x−π]\lim_{x \rightarrow \pi/6}\left\lbrack \frac{3\sin x - \sqrt{3}\cos x}{6x - \pi} \right\rbrack

A

3\sqrt{3}

B

13\frac{1}{\sqrt{3}}

C

−3- \sqrt{3}

D

−13- \frac{1}{\sqrt{3}}

Answer

13\frac{1}{\sqrt{3}}

Explanation

Solution

Using L–Hospital’s rule,

lim⁡x→π/63cos⁡x+3sin⁡x6=3.32+3.126=13\lim_{x \rightarrow \pi/6}\frac{3\cos x + \sqrt{3}\sin x}{6} = \frac{3.\frac{\sqrt{3}}{2} + \sqrt{3}.\frac{1}{2}}{6} = \frac{1}{\sqrt{3}}.