Limit involving regularized lower incomplete gamma function

Limit involving regularized lower incomplete gamma function

In the course of my research I encountered the following limit:
$$\tag{1}\lim_{n\rightarrow\infty}\frac{\gamma\left[\frac{n}{2},\frac{1}{2}\left(\frac{c}{c+f(n)/\sqrt{n}}\right)\left(n+\sqrt{-4n\ln\left(2-2b^{1/g(n)}\right)}\right)\right]}{\Gamma(n/2)}$$
where $\gamma(s,x)=\int_0^xt^{s-1}e^{-t}dt$ is the lower incomplete gamma
function which is regularized via division by gamma function
$\Gamma(n/2)$, $c>0$ and $1/2<b<1$ are constants, function $g(n)$ is
positive and asymptotically greater than $n$ (i.e. $g(n)=o(n)$).
I am interested in the behavior of this limit in terms of the positive
function $f(n)$. My conjecture is as follows:
If $f(n)$ is asymptotically upper bounded by $\sqrt{\ln n}$, i.e.
$f(n)=\mathcal{O}(\sqrt{\ln n})$, then the limit in (1) is unity;
If $f(n)$ is asymptotically greater than $\sqrt{\ln n}$, i.e.
$f(n)=\omega(\sqrt{\ln n})$, then the limit in (1) is zero.
This is supported via numerical analysis using MATLAB, but I can't prove
it analytically. Can anyone help?