Question

${lim}_{x\to 0}\frac{cosecx-cotx}{x}$

Answer 1

${lim}_{x\to 0}\frac{cosecx-cotx}{x}$

$={lim}_{x\to 0}(\frac{1}{sinx}-\frac{cosx}{sinx})\times \frac{1}{x}={lim}_{x\to 0}\left(\frac{1}{sinx}\left(\frac{1-cosx}{x}\right)\right)$

$={lim}_{x\to 0}\left(\frac{1}{sinx}\left(\frac{2si{n}^2\frac{x}{2}}{x}\right)\right)$

$={lim}_{x\to 0}\frac{2}{\frac{sin}{x}}\frac{si{n}^2\frac{x}{2}}{x^2}$

$=2\frac{1}{{lim}_{x\to 0}\frac{sin}{x}}\times {lim}_{x\to 0}{\left(\frac{sin\frac{x}{2}}{\frac{x}{2}}\right)}^2\times \frac{1}{4}$

$=2\frac{1}{\left(1\right)}\times 1\frac{1}{4}=\frac{1}{2}$