Question

${lim}_{x\to 0}\frac{x^{3}cotx}{1-cosx}$

Answer 1

${lim}_{x\to 0}\frac{x^{3}cotx}{1-cosx}$

$={lim}_{x\to 0}\frac{x^3}{tanx(1-cosx)}$

$={lim}_{x\to 0}\frac{x^3}{tanx.2si{n}^2\frac{x}{2}}$

$={lim}_{x\to 0}\frac{1}{\frac{tanx}{x}\times \frac{2si{n}^2\frac{x}{2}}{x^2}}$

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

$=\frac{1}{1\times 2\times 1\times \frac{1}{4}}[\because {lim}_{0\to 0}\frac{sin\theta }{\theta }=1,{lim}_{0\to 0}\frac{tan\theta }{\theta }=1]$

=2