Limit involving DiracDelta gives zero [closed]

Question

Why is the following limit zero?

In[147]:= Limit[DiracDelta[ϵ], ϵ -> 0]

Out[147]= 0

It does not matter if the limit is taken from either sides (+/-), since the Delta function is even. The Dirac Delta function is usually defined to be infinite at zero argument.

\begin{equation}\delta(x) = \begin{cases} +\infty, & x = 0 \\ 0, & x \ne 0 \end{cases}\end{equation}

In Mathematica they choose not to define DiracDelta[0], but why does this limit give zero?

Answer 1

(Moved from the comments since I guess this is an answer...)

Because the limit is 0. Heuristically, if you zoom into the function near $x=0$, it is of course just a flat line $\delta(x)=0$, so by the definition of the limit (which doesn't depend the actual function value at $x=0$) it gives 0.