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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?

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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.

One often encounters functions that are equal to their Taylor expansion only on a small interval (disc if the function is complex), or only at one point (the point around which the expansion was made). Such functions have their place in physics. For example, to express that property of a signal that its intensity at time B is not determined by its properties at a different time A, on the interval A-B the signal has to be described by function whose Taylor expansion around A cannot reproduce its value at B.

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