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 $$\lim\Big(\frac{e^{-ah}+e^{-bh}}2\Big)^n =\lim\exp\Big(-\frac{a+b}2\,nh\Big) \\ =\exp\Big(-\frac{a+b}2\,c\Big).$$