I'm preparing for an exam and trying to make some sense of the growth of the different exponential functions. I picked the trickiest functions for myself and tried to sort them according to the big O notation. Since it's hard to use L' Hospital's rule on exponential functions I at least want to get a feeling for the order of growth.
$$ n^2 = O(ln(n) ^ {ln(n)}) = O(n^{ln(n)}) = O(2^{\sqrt{n}})=O(2^{(n/2)}) = O(n ^{\sqrt(n)}) = O(n!) = O(n ^n) $$
I'm particularly unsure about the right place of fac(n). Can anyone verify if that is correct?