Limited edition

Calculus Level 4

$\large \lim_{\alpha\rightarrow 0}\frac{e^{\cos(\alpha^{n})}-e}{\alpha^{m}} = -\frac{e}{2}$

Let $m$ and $n$ be two constant positive integers greater than 1 such that the equation above holds true.

Find the ratio $\frac{m}{n}$ .