I am trying to find a demonstration to find the exact order with respect to the order of growth between the following functions:
$2^n-a^n-n^n-n!$
I first know that the smallest function expression will probably be n!, due to it is a product which factors are decreasing $n!=n * (n-1) * (n-2) * ... * 2 * 1$
However, in order to compute the difference or the order of growth between $2^n, a^n,n^n$ I have tried to compute the $\lim\limits_{n\to\infty} \frac{n^n}{a^n}$ and $\lim\limits_{n\to\infty} \frac{a^n}{2^n}$ but I cannot solve it.