# Calculate limit using taylor series

## Homework Statement

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

## 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