Question

${lim}_{x\to 0}\frac{3si{n}^{2}x-2sin{x}^2}{3x^2}$

Answer 1

${lim}_{x\to 0}\frac{3si{n}^{2}x-2sin{x}^2}{3x^2}$

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

$={lim}_{x\to 0}{\left(\frac{sinx}{x}\right)}^2-\frac{2}{3}{lim}_{x\to 0}\frac{sin{x}^2}{x^2}$

$=1-\frac{2}{3}\times 1[\because {lim}_{\theta \to 0}\frac{sin\theta }{\theta }=1]$

$=1-\frac{2}{3}=\frac{3-2}{3}=\frac{1}{3}$