# Behavior of $\lim_{n\rightarrow\infty}\left(\frac{2\sqrt{a(a+b/(\sqrt{n}+\epsilon))}}{2a+b/(\sqrt{n}+\epsilon)}\right)^{\frac{n}{2}}$

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: $0<b<a<\infty$, and $\epsilon\in\mathbb{R}$. I am interested in the following:

1. 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?
2. 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 $b$?
3. what happens to this limit when $\epsilon=0$?
-

Claim: The limit is $\exp(-b^2/(16a^2))$, irrespective of $\epsilon$.
Proof: Let $x=b/(2a(\sqrt{n}+\epsilon))$, then one asks for the behaviour of $$K_n=\left(\frac{1+2x}{(1+x)^2}\right)^{n/4}$$ when $n\to\infty$, with $x\to0$. But $\frac{1+2x}{(1+x)^2}=1-\frac{x^2}{(1+x)^2}=1-x^2+o(x^2)$ and $x^2\sim c^2/n$ with $c=b/(2a)$, hence $$K_n=\left(1-\frac{c^2}n+o\left(\frac1n\right)\right)^{n/4}\longrightarrow\exp\left(-\frac{c^2}4\right).$$