A question about Fermi-Dirac Distribution function

It seems more like a mathematical question, about the property of Fermi-Dirac Distribution function $$f=\frac{1}{e^{(E-\mu)/k_BT}+1}$$ where $\mu$ is the chemical potential and $k_B$ is the Boltzmann constant.

I find that $\frac{\partial^nf}{\partial T^n}|_{T\to0}=0$, for any positive integer $n$. That is true for either $T\to0^+$ or $T\to0^-$.

This seems that we are unable to taylor expand $f$ near $T=0$. Or, say, we are unable to use any function of $T$ to approximate the Fermi-Dirac Function according to the order of T near $T=0$ point.

Are there any physical meaning or application of this property? Why nature gives this property to the widely used Fermi-Dirac Function?

The property you've found (Taylor expansion $\neq$ original function on any interval around the point) has nothing to do with physics or nature, and is not particularly connected only to the Fermi-Dirac distribution. The failure may be limited only to some special points; if some expansion is really needed regardless of this, often you can expand the function around some other point, in the present case, say $T= 1~$K.