Limit of (arcsen(2x)-2arcsen(x))/x^3 as x->0

[tsuwal]

Homework Statement

Limit of (arcsen(2x)-2arcsen(x))/x^3 as x->0

Homework Equations

arsen(x)'=1/Sqrt(1-x^2)

The Attempt at a Solution

It's an 0/0 indeterminancy, to solve it, I had to use L'hopital's rule once simplify de expression, multiplying by the conjugate root. Is there an easier way?

 

[mfb]

As you don't get rid of the trigonometric functions without l'Hospital, I think this ok.

$$\frac{\frac{2}{\sqrt{1-(2x)^2}}-\frac{2}{\sqrt{1-(x)^2}}}{x^2} = 2 \frac{1}{\frac{1}{\sqrt{1-(2x)^2}}+\frac{1}{\sqrt{1-(x)^2}}} \frac{\frac{1}{1-(2x)^2}-\frac{1}{1-(x)^2}}{x^2} \to \frac{2}{2} \frac{\frac{1}{1-(2x)^2}-\frac{1}{1-(x)^2}}{x^2} \to \dots$$

 

[tsuwal]

Thanks!