Fermi-Dirac distribution at T->0 and \mu->\epsilon_0

[mcas]

Homework Statement

Starting with F-C distrubution for $T>0$
$$f(\epsilon_\vec{k})=(e^{\frac{(\epsilon_\vec{k} - \mu)}{kT}}+1)^{-1}$$
derive a distrubution at limit of $T->0$ when $\mu(T)-> \epsilon_F$

Relevant Equations

$f(\epsilon_\vec{k})=(e^{\frac{(\epsilon_\vec{k} - \mu)}{kT}}+1)^(-1)$
$\mu(T=0)=\epsilon_F$

The limit itself is pretty easy to calculate
$lim_{T->0} \ lim_{\mu->\epsilon_F} \ (e^{\frac{(\epsilon_F - \mu)}{kT}}+1)^{-1} = \frac{1}{2}$

But I'm very confused about changing $\epsilon_\vec{k}$ to $\epsilon_F$. Why do we do this?

 

[anuttarasammyak]

Depending on $\epsilon_k$ with comparison to Fermi energy as T $\rightarrow$ 0,
For $\epsilon_k > \epsilon_f$ $f \rightarrow ?$
For $\epsilon_k < \epsilon_f$ $f \rightarrow ?$
and
For $\epsilon_k = \epsilon_f$ $f = 1/2$ for any temperature $T \neq 0$.