In one of its lower paragraphs, Wikipedia describes the dirac delta distribution as the limit of a sequence of zero-centered normal distributions.
https://en.wikipedia.org/wiki/Dirac_delta_function
I don't know much about "distributions" per se, but, are there other integrals of sequences of functions that converge to the dirac delta distribution? Or is this sequence unique in some way?
You can try to prove the following yourself: Let $\chi$ be a continuous nonnegative function such that $$ \int_{-\infty}^{\infty} \chi(x)\,dx = 1 $$ and for each $\delta > 0$, let $$ \chi_\delta(x) = \delta^{-1}\chi(\delta^{-1}x). $$ Then for any bounded continuous function $f$ on the real line, $$ \lim_{\delta\rightarrow 0} \int_{-\infty}^{\infty} \chi_\delta(x)f(x)\,dx = f(0). $$