I want to compute the limit of the following fucntion when $n\to\infty$:$$f(n)=\frac{2 n \left(1-\frac{1}{n}\right)^{g(n)}}{2-3 n \ln \left(1-\left(1-\frac{1}{n}\right)^{g(n)}\right)}$$
It depends on a function $g(n)$ which can have several asymptotic bounds, depending on which the limit will result in a different value:$$\lim_{n\to\infty} f(n)=0 \quad[O(g(n))\subset O(n)]$$$$\lim_{n\to\infty} f(n)=\frac{2}{3e - 3e \ln(e-1)} \quad[g(n)=\Theta(n)]$$$$\lim_{n\to\infty} f(n)=\frac{1}{\frac{3}{2}+e^\gamma} \quad[g(n)=\Theta(nH_n)]$$As seen, when $g(n)$ is asymptotically smaller than $n$, it tends to 0. In case it grows exactly as $n$ does, it converges in a real value, as well as in the $g(n)=\Theta(nH_n)$ case.
$$\lim_{n\to\infty} f(n)=\frac{2}{3}\enspace[g(n)=\Theta(n\ln(\ln(n)))];\enspace \lim_{n\to\infty} f(n)=\frac{2}{3}\quad[g(n)=\Theta(n\ln(\sqrt{n}))]$$As $g(n)$ can't exceed the $\Theta(nH_n)$ bound, I'm interested in knowing if between $O(n)$ and $O(nH_n)$ the limit converges to the same value for all functions within such range. I tried with the above 2 and it seems the limit converges to $\frac{2}{3}$, but I didn't find a way to prove it for all functions of the set $O(nH_n)-O(n)$.
Also, is there any way to prove the $O(g(n))\subset O(n)$ case for all functions of the set $O(n)$?