I am having trouble expressing the behavior of the following limit:
$$\lim_{n\rightarrow\infty}\left(\frac{2\sqrt{a(a+b/n^{0.5-\epsilon})}}{2a+b/n^{0.5-\epsilon}}\right)^{\frac{n}{2}}$$
After some simple arithmetic manipulations I can simplify this expression to this:
$$\lim_{n\rightarrow\infty}\left(1+\frac{b^2n^{-1+2\epsilon}}{4a^2+4abn^{-0.5+\epsilon}}\right)^{-\frac{n}{4}}$$
with the following constraints on the parameters: $0my previous and related question.
I am perplexed on how to actually prove the statements for $0<\epsilon<0.5$ and $-0.5<\epsilon<0$. It'd be great if someone could help!