In my study of statistical physics I have found a rather embarrassing issue. Classical thermodynamics is usually recovered in the limit $T\to +\infty$, which corresponds to $\beta=\frac{1}{k_B T}\to 0$. But oftentimes I see the classical limit written as $z\to 0$, where $z=e^{\beta\mu_0}$ is the fugacity.
Shouldn't we have $z\to 1$ instead?
It depends on what you are interested in. It is often more physically relevant to think at fixed number of particles rather than fixed $\mu$, so $\mu$ picks up an implicit temperature dependence.
Take for example classical particles following Maxwell-Boltzmann statistics having the same energy $\epsilon$. You have: $$ n = z e^{-\beta\epsilon} $$
So when $\beta\to0$, to maintain the same number of particles:
$$ z\to n \\ \mu \to T\ln n $$
Similarly for Bose-Einstein statistics you have:
$$ n = \frac{ze^{-\beta\epsilon}}{1-ze^{-\beta\epsilon}} $$
so when $\beta\to0$, to maintain the same number of particles:
$$ z\to \frac{n}{n+1} \\ \mu \to T\ln\left(\frac{n}{1+n}\right) $$
and for Fermi-Dirac statistics you have:
$$ n = \frac{ze^{-\beta\epsilon}}{1+ze^{-\beta\epsilon}} $$
so when $\beta\to0$, to maintain the same number of particles:
$$ z\to \frac{n}{1-n} \\ \mu \to T\ln\left(\frac{n}{1-n}\right) $$
The idea is that these formulas all give the same limit when $n\ll1$ ie the low density limit, which gives $z\to 0$. Note that the order in which you take the limit $\beta\to0$ and $n\to 0$ matters.
Hope this helps.
