calculus and analysis - Calculating a limit with a result that is discontinuous in the parameters

The following limit is left unevaluated (Edit: added the assumption that $\epsilon$ is real thanks to the comment below):

Limit[ 1/(1 + Exp[ϵ/T]), T->0, Direction->-1, Assumptions->ϵ ∈ Reals]

However, the limit is easy to calculate (it is the Fermi-Dirac distribution), and it gives $\theta(-\epsilon)$. It seems that MMA has a hard time when the result is discontinuous in the parameters.
Nevertheless, MMA can actually calculate the limit, if we help it by explicitly splitting up the parameter space:

In[2]:= Limit[1/(1 + Exp[ϵ/T]), T->0, Direction->-1, Assumptions->#] & /@ {ϵ > 0, ϵ < 0}

Out[2]= {0, 1}

Is there a way to make MMA automatically calculate such limits, without explicitly splitting up the parameter space?