Let m and n be two positive integers greater than 1. If  $\underset{\alpha \to 0}{\lim }\frac{e^{\cos \left({\alpha }^n\right)}-e}{{\alpha }^m}=-\frac{e}{2}$ then the value of  $\frac{m}{n}$is

$\underset{\alpha \to 0}{Lt}\frac{e(e^{\cos {\alpha }^n-1}-1)}{{\alpha }^m}=\frac{-e}{2}$

$=\underset{\alpha \to 0}{Lt}\frac{e^{\cos {\alpha }^n-1}-1}{\cos {\alpha }^n-1}\underset{\alpha \to 0}{Lt}\frac{\cos {\alpha }^n-1}{{\alpha }^m}=\frac{-1}{2}$

$=\underset{\alpha \to 0}{Lt}\frac{1-\cos {\alpha }^n}{{\alpha }^m}=\frac{1}{2}$, this is true for  $n=1,m=2$

$\frac{m}{n}=\frac{2}{1}=2$