Question:
Evaluate the following limits:
$\lim _{x \rightarrow \frac{\pi}{6}} \frac{\cot ^{2} x-3}{\operatorname{cosec} x-2}$
Solution:
$=\lim _{x \rightarrow \frac{\pi}{6}} \frac{\cot x \times \cot x-3}{\csc x-2}$
$=\lim _{x \rightarrow \frac{\pi}{6}} \frac{(\cos x \times \cos x)-3 \times \sin x \times \sin x}{\sin x(1-2 \sin x)}$
$=\lim _{x \rightarrow \frac{\pi}{6}} \frac{1-4 \times \sin x \times \sin x}{\sin x(1-2 \sin x)}$
$=\lim _{x \rightarrow \frac{\pi}{6}} \frac{(1-2 \sin x) \times(1+2 \sin x)}{\sin x(1-2 \sin x)}$
$=4$
$\therefore \lim _{x \rightarrow \frac{\pi}{6}} \frac{\cot x \times \cot x-3}{\csc x-2}=4$
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