# Find the limit without using Maclaurin series

Find a limit, $$\lim_{x\to 0} \frac{1-\cos{(1-\cos{(1-\cos x)})}}{x^8}$$ without using Maclaurin series. My attempt was to use L'hopital's rule but that's just too much, and chances of not making a mistake trough repetitive differentiation are very low.

Hint: Use $1-\cos t = 2\sin^2 \frac{t}{2}$ and the chain rule for limit. (ask if you need further hint)