Question

Evaluate:

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

Answer 1

${lim}_{x\to \frac{\pi }{6}}\frac{{\cot }^{2}x-3}{\csc x-2}$ $(\frac{0}{0}\to \text{indeterminate form})$

$={lim}_{x\to \frac{\pi }{6}}\frac{{\csc }^{2}x-1-3}{\csc x-2}$

$[\because {\cot }^{2}x={\csc }^{2}x-1]$

$={lim}_{x\to \frac{\pi }{6}}\frac{{\csc }^{2}x-4}{\csc x-2}$

$={lim}_{x\to \frac{\pi }{6}}\frac{(\csc x+2)(\csc x-2)}{(\csc x-2)}$

$={lim}_{x\to \frac{\pi }{6}}(\csc x+2)$

$=\csc \frac{\pi }{6}+2$

$=2+2$

$=4$