Does Mathematica interprets representations of Dirac's Delta?

Question

Hello I'm trying to take the limit of this expression in Mathematica

I know this is a representation of Dirac's Delta (more precisely pi times Delta), but when I do this, the answer is 0.

So my question is This happens with every representation of every "treaky" function?

I have to take the limit of a more awful expression and if I can't fix this, then I won't try with that.

Thanks