Evaluate the following limits:

$=\lim _{x \rightarrow \frac{\pi}{6}} \frac{2 \times \sin x \times \sin x+\sin x-1}{2 \times \sin x \times \sin x-3 \sin x+1}$

$=\lim _{x \rightarrow \frac{\pi}{6}} \frac{(2 \sin x-1) \times(\sin x+1)}{(2 \sin x-1)(\sin x-1)}$

$=-3$

$\therefore \lim _{x \rightarrow \frac{\pi}{6}} \frac{2 \times \sin x \times \sin x+\sin x-1}{2 \times \sin x \times \sin x-3 \sin x+1}=-3$