Question:
Evaluate the following limits:
$\lim _{x \rightarrow \frac{\pi}{6}} \frac{2 \sin ^{2} x+\sin x-1}{2 \sin ^{2} x-3 \sin x+1}$
Solution:
$=\lim _{x \rightarrow \frac{\pi}{6}} \frac{2 \times \sin x \times \sin x+\sin x-1}{2 \times \sin x \times \sin x-3 \sin x+1}$
$=\lim _{x \rightarrow \frac{\pi}{6}} \frac{(2 \sin x-1) \times(\sin x+1)}{(2 \sin x-1)(\sin x-1)}$
$=-3$
$\therefore \lim _{x \rightarrow \frac{\pi}{6}} \frac{2 \times \sin x \times \sin x+\sin x-1}{2 \times \sin x \times \sin x-3 \sin x+1}=-3$
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