Question

${lim}_{x\to 0}\frac{xcosx+2sinx}{x^2+tanx}$

Answer 1

${lim}_{x\to 0}\frac{xcosx+2sinx}{x^2+tanx}$

Dividing numerator and denominator by x

$={lim}_{x\to 0}\frac{cosx+\frac{2sinx}{x}}{x+\frac{tanx}{x}}$

$=\frac{{lim}_{x\to 0}cosx+li{m}_{x\to 0}\frac{2sinx}{x}}{{lim}_{x\to 0}x+{lim}_{x\to 0}\frac{tanx}{x}}$

$=\frac{cos0+2{lim}_{x\to 0}\frac{sinx}{x}}{x+{lim}_{x\to 0}\frac{tanx}{x}}$

$=\frac{1+2}{0+1}=3[\because {lim}_{x\to 0}\frac{sinx}{x}=1,{lim}_{x\to 0}\frac{tanx}{x}=1]$

=3