Question

${lim}_{x\to 0}\frac{2sinx-sin2x}{x^3}$

Answer 1

${lim}_{x\to 0}\frac{2sinx-sin2x}{x^3}$

$={lim}_{x\to 0}\frac{2sinx-2sinxcosx}{x^3}$

$={lim}_{x\to 0}\frac{2sinx(1-cosx)}{x^3}$

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

$={lim}_{x\to 0}\frac{sinx\times si{n}^2\frac{x}{2}}{x\times \frac{x^4}{4}}$

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

$=1\times 1({lim}_{\theta \to 0}\frac{sin\theta }{\theta }=1)$

=1