Question

Evaluate $\underset{x\to 0}{lim}\frac{1-\cos 3x}{x^2}$.

Answer 1

Let $L=\underset{x\to 0}{lim}\frac{1-\cos 3x}{x^2}$

The expression takes the form $\frac{0}{0}$.

Using L-Hospital Rule,

$L=\underset{x\to 0}{lim}\frac{0+3\sin 3x}{2x}$

Again, the expression takes the form $\frac{0}{0}$.

Using L-Hospital Rule,

$L=\underset{x\to 0}{lim}\frac{9\cos 3x}{2}$

$=\frac{9}{2}\cos 0^o$

$=\frac{9}{2}$