Calculate limit using taylor series

1. Sep 10, 2012

Hernaner28

1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. 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

2. Sep 11, 2012

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.