Question

${lim}_{x\to 0}\frac{xtanx}{1-cos2x}$

Answer 1

${lim}_{x\to 0}\frac{xtanx}{1-cos2x}$

$={lim}_{x\to 0}\frac{xtanx}{2si{n}^{2}x}$ [Dividing numerator and denominator by $x^2$]

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

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

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

$=\frac{1}{2\times 1}=\frac{1}{2}$