Question

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

Answer 1

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

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

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

$={lim}_{x\to 0}1\times \frac{x}{2}$