I'm having a hard time characterising the behavior of the following expression:
$\lim_{n\rightarrow\infty}\left(\frac{2\sqrt{a(a+b/(\sqrt{n}+\epsilon))}}{2a+b/(\sqrt{n}+\epsilon)}\right)^{\frac{n}{2}}$
with the following constraints on the parameters: $0, and $\epsilon\in\mathbb{R}$. I am interested in the following:
- for $\epsilon>0$, does this limit go to zero or does it go to some constant $C$? If it can both go to zero or to some constant $C>0$, what are the conditions on the value of $\epsilon$ as a function of $a$ and $b$ which leads to these outcomes, if any?
- for $\epsilon<0$, does it always go to some constant $C<1$, or can it go to 1 for some $\epsilon$, if it's a function of $a$ and $b$?
- what happens to this limit when $\epsilon=0$?