Limits Continuity And Differentiability

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