Question

Find ${lim}_{x\to 0}\frac{\mathrm{asin}x-\sin 2x}{\tan 3x}$

Answer 1

${lim}_{x\to 0}\frac{a\sin x-\sin 2x}{\tan 3x}$

$={lim}_{x\to 0}\frac{a\frac{\sin x}{x}\times x-\frac{\sin 2x}{2x}\times 2x}{\frac{\tan 3x}{3x}\times 3x}$

$={lim}_{x\to 0}\frac{a\times 1\times x-1\times 2x}{1\times 3x}$ , since ${lim}_{\theta \to 0}\frac{\sin \theta }{\theta }=1,{lim}_{\theta \to 0}\frac{\tan \theta }{\theta }=1$

$={lim}_{x\to 0}\frac{ax-2x}{3x}$

$={lim}_{x\to 0}\frac{(a-2)x}{3x}$

$={lim}_{x\to 0}\frac{(a-2)}{3}$

$=\frac{a-2}{3}$