$\lim _{x \rightarrow \frac{\pi}{6}} \frac{\cot ^{2} x-3}{\operatorname{cosec} x-2}$

solution:

Given $\lim _{x \rightarrow \frac{\pi}{6}} \frac{\cot ^{2} x-3}{\operatorname{cosec} x-2}$

We know that

$\cot ^{2} x=\operatorname{cosec}^{2} x-1$

By using this in given equation we get

$\Rightarrow$ $\lim _{x \rightarrow \frac{\pi}{4}} \frac{\left(\operatorname{cosec}^{2} x-1\right)-3}{\operatorname{cosec} x-2}=\lim _{x \rightarrow \frac{\pi}{4}} \frac{\operatorname{cosec}^{2} x-4}{\operatorname{cosec} x-2}$

Again using $a^{2}-b^{2}$ identity the above equation can be written as

$\Rightarrow$$\lim _{x \rightarrow \frac{\pi}{6}} \frac{\operatorname{cosec}^{2} x-4}{\operatorname{cosec} x-2}=\lim _{x \rightarrow \frac{\pi}{6}} \frac{(\operatorname{cosec} x-2)(\operatorname{cosec} x+2)}{\operatorname{cosec} x-2}$

On simplification and applying the limits we get

$\Rightarrow$$\lim _{x \rightarrow \frac{\pi}{6}} \frac{(\operatorname{cosec} x-2)(\operatorname{cosecx}+2)}{\operatorname{cosec} x-2}=\lim _{x \rightarrow \frac{\pi}{6}}(\operatorname{cosecx}+2)=2+2=4$

$\Rightarrow$$\lim _{x \rightarrow \frac{-1}{6}} \frac{\cot ^{2} x-3}{\operatorname{cosec} x-2}=4$