If a sequence of random variables $X_n$ are asymptotically Normal, in what sense is $e^{X_n}$ asymptotically Normal as well?

Question

If a sequence of random variables $X_n$ are asymptotically Normal, in what sense is $e^{X_n}$ asymptotically Normal as well?

I read the following inside a book:

Is it surprising that if $X_n$ is approximately Normal for $n$ large, then $e^{X_n}$ is also approximately Normal for $n$ large? Asymptotically, exponential functions are linear. But then again, asymptotically any two people have the same age, and this is even a better approximation since they differ only by a constant. John Maynard Keynes said “In the long run we are all dead”; in the short run, should we be happy to be asymptotically right but exponentially wrong?

I am confused what the author is trying to say here. Is he trying to say that while $e^{X_n}$ is also approximately Normal for $n$ large, it is a bad approximation? Thanks.

Answer 1

Lemma: If $X_n\sim AN(\mu,\sigma^2)$ then $g(X_n)\sim AN(g(\mu),\sigma^2(g'(\mu))^2)$ for all "nice" $g$.

Your $g(t)=e^t$ here. The proof of the lemma is short using Taylor Expansion of $g(X_n)$ around $g(\mu)$. Can you complete it?