This limit is not a usual limit, but the limit in the weak topology. Such limits are not imlemented in current CASes. All that is not a simple matter. See that Wiki article and/or Encyclopedia of Mathematics as a first reading. Something similar: an atom cannot be represented as a very small ball.

What will you be using DiracDelta[] for?

I'm not using DiracDelta for anything, I just want to represent it.

@LeonidShifrin: The integral of the type $\int_{-\infty}^\infty\delta(x-y)f(y)\,dy$ makes no sense in math (see Generalized function to this end). That was noticed many times at the forum.

@user64494 You can always use the regularization and the limiting procedure to make sense of these kinds of integrals, understood as a limiting procedure rather than directly as a standard (Riemann) integral. From the practical viewpoint, what matters is the order of the limits, which is what I mentioned in my comment.

@LeonidShifrin: I prefer arguments over a tonn of emotional and ungrounded words (" From the practical viewpoint" etc).

@user64494 What I gave you is an argument. An (Riemann) integral is defined as a limiting procedure. You have two limits to take then. As long as we understand what the notation of delta function means - namely, a specific order of limits, integral first, and regularization last, there is no problem with the validity of the integral. I have spent a big chunk of my life working with delta functions in physics, first studying, then getting and publishing actual results. What I told you was what mattered in practice in my work. And yes, I am familiar with formal theory of generalized functions.

@LeonidShifrin you're feeding a troll. user64494 says "[...] That was noticed many times at the forum." Yes, by you and you alone. You spend so much time arguing against other people's use of the Dirac δ-function and yet refuse to study how it is actually used every day by millions of physicists/scientists. Has it occurred to you that maybe it's you who's mistaken here and in the dozens of other posts you complain about regularly?

The paragraph en.wikipedia.org/wiki/Dirac_delta_function#As_a_measure might be useful. It's an abuse of notation that is to be understood in the context of Lebesgue integrals wrt the Dirac measure (and not the Lebesgue measure $dx$, and that's the abuse).

Does this answer your question? How to deal with a Dirac delta function numerically?

@LeonidShifrin : I looked at your current 672 answers at this forum and found only that on the topic.

@anderstood: The interpretation of the $\delta$-distribution as an atomic measure causes problems with ODEs of such type $y''(x)+y(x)=\delta(x-1)$.

@Roman "you're feeding a troll" - apparently yes, at least in the context of this Q/A. I was confused by their high reputation.

Colleagues, let us discuss topics, not persons. Do you understand me?

Are you talking about a graphical representation? A common convention is to use an upward pointing arrow. See en.wikipedia.org/wiki/Dirac_delta_function#/media/….