Question

Solve: $\underset{x\to \pi /4}{lim}\frac{{\csc }^{2}x-2}{\cot x-1}$

Answer 1

${lim}_{x\to \frac{\pi }{4}}\frac{{\csc }^{2}x-2}{\cot x-1}$

$={lim}_{x\to \frac{\pi }{4}}\frac{{\csc }^{2}x-1-1}{\cot x-1}$

$={lim}_{x\to \frac{\pi }{4}}\frac{{\cot }^{2}x-1}{\cot x-1}$

$={lim}_{x\to \frac{\pi }{4}}\frac{(\cot x-1)(\cot x+1)}{\cot x-1}$

$={lim}_{x\to \frac{\pi }{4}}\cot x+1$

$=\cot \frac{\pi }{4}+1$

$=1+1=2$