I am having some difficulty with multivariable asymptotics. Let me provide a concrete example of the kind of thing I mean.
Stirling's approximation for $n!$ is
$ n! \sim \sqrt{2 \pi n}\left( \frac{n}{e} \right)^n. $
For ease of notation, denote the right-hand side of the above by $S(n)$. The precise meaning of $\sim$ is that
$ \lim_{n \rightarrow \infty} \frac{n!}{S(n)} = 1. $
Suppose now we have a new variable $k$ and want to ask about the asymptotics of $(n!)^k$ where $k$ is allowed to grow with $n$. A proposed asymptotic expression may not (likely does not) exist if $n$ and $k$ are allowed to grow at arbitrary rates. Instead, I would like to view $k$ as a function $k(n)$ and determine the largest possible order of growth for $k(n)$ for which a proposed asymptotic formula will hold. A way to begin might be to ask what is the largest possible order of growth for which
$ \lim_{n \rightarrow \infty} \frac{(n!)^{k(n)}}{(S(n))^{k(n)}} = 1. $
How might one approach such a problem?