Question

Find the limit :
$\underset{x\to \frac{\pi }{6}}{lim}\frac{{\cot }^{2}x-3}{cosecx-2}$

Answer 1

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

Apply L' Hospital Rule

$\underset{x\to \frac{\pi }{6}}{lim}=\frac{2\cot x\left({\csc }^{2}x\right)}{\csc x\cot x}$

$\underset{x\to \frac{\pi }{6}}{lim}=2\csc x$

$2\times 2$

$=4$