$\begin{aligned} & \text { } \lim _{x \rightarrow \frac{\pi}{6}} \frac{3 \sin x-\sqrt{3} \cos x}{6 x-\pi} \\ & \lim _{x \rightarrow \frac{\pi}{6}} \frac{3 \cos x+\sqrt{3} \sin x}{6}=\frac{3 \times \frac{\sqrt{3}}{2}+\sqrt{3} \times \frac{1}{2}}{6} \\ & =\frac{4 \sqrt{3}}{2 \times 6}=\frac{1}{\sqrt{3}}\end{aligned}$