The value of $\underset{x→0}{\lim}\frac{1-\cos(1-\cos x)}{x^4}$ is:

$\frac{1}{8}$

$\underset{x→0}{\lim}\frac{1-\cos(1-\cos x)}{x^4}=\underset{x→0}{\lim}\frac{1-\cos(2\sin^2x/2)}{x^4}=\underset{x→0}{\lim}\frac{2\sin^2(\sin^2x/2)}{x^4}$

$=\underset{x→0}{\lim}2(\frac{\sin(\sin^2x/2)}{\sin^2x/2}.\frac{\sin^2x/2}{x^2/4}.\frac{1}{4})^2=1/8$