Question

Evaluate $\underset{x\to 0}{lim}\frac{3\sin x-\sin 3x}{x^2}$

Answer 1

$\underset{x\to 0}{Lt}\frac{3\sin x-\sin 3x}{x^2}$

$\underset{x\to 0}{Lt}\frac{3\sin x-3\sin x+4{\sin }^{3}x}{x^2}$ $[\sin 3x=3\sin x-4{\sin }^{3}x]$

$=\underset{x\to 0}{Lt}4{\left(\frac{\sin x}{x}\right)}^2\sin x$

$\underset{x\to 0}{Lt}\frac{\sin x}{x}=1$

$=\underset{x\to 0}{Lt}4\sin x=4\sin 0=0$.