_ x 6 3 x-3 x 6 x- equals - Vidyadip

MCQ

$\lim _{x \rightarrow \frac{\pi}{6}} \frac{3 \sin x-\sqrt{3} \cos x}{6 x-\pi}$ equals

A
$\sqrt{3}$
✓
$\frac{1}{\sqrt{3}}$
C
$-\sqrt{3}$
D
$-\frac{1}{\sqrt{3}}$

✓

Answer

Correct option: B.

$\frac{1}{\sqrt{3}}$

(B)
Applying L-Hospital's Rule, we get
$\lim _{x \rightarrow \frac{\pi}{6}} \frac{3 \sin x-\sqrt{3} \cos x}{6 x-\pi}=\lim _{x \rightarrow \frac{\pi}{6}}\left(\frac{3 \cos x+\sqrt{3} \sin x}{6}\right)$
$=\frac{3 \cdot \frac{\sqrt{3}}{2}+\sqrt{3 \cdot \frac{1}{2}}}{6}$
$=\frac{1}{\sqrt{3}}$

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