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: $0, and $-0.5<\epsilon<0.5$. For $0<\epsilon<0.5$ it seems to go to zero, and for $-0.5<\epsilon<0$ it seems to go to one (at least it looks that way when plotting it in MATLAB, see pictures for $a=1$, $b=0.1$.) At $\epsilon=0.5$ it's a constant function of $a$ and $b$, according to an answer to my previous and related question. f(n) vs. $n$ for $\epsilon=0.3$">
f(n) vs. $n$ for $\epsilon=-0.1$">
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!