\(\lim\limits_{x \to\frac{\pi}{6}}\frac{cot^2x-3}{cosec\,x-2}\)
\(=\lim\limits_{x \to\frac{\pi}{6}}\frac{cosec^2x-1-3}{cosec\,x-2}\)
\(=\lim\limits_{x \to\frac{\pi}{6}}\frac{cosec^2x-4}{cosec\,x-2}\)
\(=\lim\limits_{x \to\frac{\pi}{6}}\frac{(cosec\,x-2)(cosec\,x+2)}{(cosec\,x-2)}\)
\(=\lim\limits_{x \to\frac{\pi}{6}}(cosec\,x+2)\) [x ≠ π/6]
\(= cosec\frac{\pi}{6}+2\)
= 2 + 2
= 4
