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 \( \frac{m}{n} \).

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