Can I use the delta method with a function that depends on n to approximate the distribution of a function of the sum of iid random variables? Let $X_1, X_2,...$ be i.i.d. random variables with finite mean $\mu$ and finite variance $\sigma^2$. From the Central Limit Theorem, we know that $\sqrt{n}(\bar{X_n}-\mu)$ tends in distribution to $N(0,\sigma^2)$ as $n\rightarrow\infty$, where $\bar{X_n}=\frac{1}{n}\sum_{i=1}^nX_i$.
I'm interested in the distribution of $f(X^n)$, where $X^n$ is the sum $X^n=n\bar{X_n}$ and $f$ is continuous and differentiable. This question is relevant and does this for $f(x)=\sqrt{x}$, but I'd like a more general result. Is the following reasoning correct?
We apply the delta method with $g(x)=f(nx)$ to see that $\frac{(f(X^n)-f(n\mu))}{\sqrt{n}f'(n\mu)}$ converges in distribution to $N(0,\sigma^2)$. (Is $g$ allowed to depend on $n$ like this?)
For sufficiently large $n$ then, $f(X^n)$ approximately follows a normal distribution $N(f(n\mu),nf'(n\mu)^2\sigma^2)$.
On the one hand, I get the same result as the question linked above for $f(x)=\sqrt{x}$, which is promising. But something slightly weird happens for $f(x)=\log(x)$. Then $g'(\mu)=\frac{n}{n\mu}=\frac{1}{\mu}$ and so $\sqrt{n}(\log(X^n)-\log(n\mu))$ converges in distribution to $N(0,\frac{\sigma^2}{\mu^2})$. But then if I want to look at the approximation of $\log(X^n)$ for large $n$, I get $N(\log(n\mu),\frac{\sigma^2}{n\mu^2})$, which has variance that decreases as $n$ increases. That seems weird but I guess it makes sense. It definitely makes sense when $f$ is bounded, as $f(X^n)$ will just approach that bound and the variance should drop to $0$. For $\log(X^n)$, the mean is still moving up but the distribution just get more narrowly concentrated around it, since $\log(X^n)$ grows so slowly for large $n$. (I'm assuming for these examples that the $X_i$ are positive.)
My main question is: am I allowed to use the delta method like this?
If so, I'd also be interested to hear if my interpretation of the $\log(X^n)$ approximation is correct.

Here is a simulation that confirms your suspicions.  The variance decreases with increasing $n$.
data norm;
do n=20, 50, 100;
do sim=1 to 100000;
    do i=1 to n;
        x=rand('normal',2,3);
        output;
    end;
end;
end;
run;

proc means data=norm noprint;
by n sim;
var x;
output out=out sum(x)=sum;
run;

data out;
set out;
log_sum=log(sum);
run;

proc means data=out var;
class n;
var log_sum;
run;

