Evaluate :\(\lim\limits_{x \to \frac{\pi}{6}}\frac{cot^2x-3}{cosec\,x-

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

Evaluate :\(\lim\limits_{x \to \frac{\pi}{6}}\frac{cot^2x-3}{cosec\,x-2}\)

Answer»

\(\lim\limits_{x \to\frac{\pi}{6}}\frac{cot^2x-3}{cosec\,x-2}\)

\(=\lim\limits_{x \to\frac{\pi}{6}}\frac{cosec^2x-1-3}{cosec\,x-2}\)

\(=\lim\limits_{x \to\frac{\pi}{6}}\frac{cosec^2x-4}{cosec\,x-2}\)

\(=\lim\limits_{x \to\frac{\pi}{6}}\frac{(cosec\,x-2)(cosec\,x+2)}{(cosec\,x-2)}\)

\(=\lim\limits_{x \to\frac{\pi}{6}}(cosec\,x+2)\) [x ≠ π/6]

\(= cosec\frac{\pi}{6}+2\)

= 2 + 2

= 4

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