Question

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

Answer 1

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

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

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

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

$={lim}_{x\to 0}\frac{xcos\frac{x}{2}}{cosxsin\frac{x}{2}}={lim}_{x\to 0}\frac{1}{cosx\times \frac{tan\frac{x}{2}}{x}}$

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

$=1\times 1\times 2=2$