Question

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

Answer 1

We have,

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

This is the $\frac{0}{0}$ form.

So, apply L-hospital rule,

$\underset{x\to 0}{lim}\frac{6\sin x.\cos x-2\cos x^2.2x}{6x}$

$\underset{x\to 0}{lim}\frac{\sin x.\cos x}{x}-\frac{4x\cos x^2}{6x}$

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

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

$=\frac{1}{3}$

Hence, this is the answer.