For $h\to0$, we have 
$$\frac{e^{-ah}+e^{-bh}}2=1-\frac{a+b}2\,h+O(h^2)
=\exp\Big(-\frac{a+b}2\,h+O(h^2)\Big)$$
and hence 
$$\Big(\frac{e^{-ah}+e^{-bh}}2\Big)^n
=\exp\Big(-\frac{a+b}2\,nh+O(nh^2)\Big).$$

So, if $nh\to c\in\mathbb R$, then $nh^2\to0$ and hence 
$$\lim\Big(\frac{e^{-ah}+e^{-bh}}2\Big)^n
=\lim\exp\Big(-\frac{a+b}2\,nh\Big) \\ 
=\exp\Big(-\frac{a+b}2\,c\Big).$$