Calculate limit using taylor series

[Hernaner28]

Homework Statement

Calculate: $$ \displaystyle \underset{x\to 0}{\mathop{\lim }}\,\frac{1-\cos \left( 1-\cos x \right)}{{{x}^{4}}}$$

Homework Equations

The Attempt at a Solution

Using Taylor series I have:

$$ \displaystyle f'\left( x \right)=\sin \left( 1-\cos x \right)\sin x$$

$$\displaystyle f'\left( 0 \right)=0$$

but as you can see, now it turns very tedious to continue differentiating. Do I have to keep differentiating until I get something? Or is there any quicker way to compute this?

Thanks

 

[clamtrox]

There is. All you need to use is the Taylor series for cos(x). Also remember that since you're taking limit of x->0, you only need to carry with you the terms of the lowest non-trivial order.