Deriving the Dirac delta function sifting property from a top hat function

Question

I am trying to derive the Dirac delta function from a top hat function whose width tends to $0$ and has an area of $1$. Here is my working below and let me know if there is anything wrong with it:

Answer 1

We want to prove $\lim_{a\to0^+}\int_{q-a}^{q+a}\frac{1}{2a}f(x)dx=f(q)$ (provided $f$ is continuous at $q$; the last $=$ in this answer uses that). Before trying, let's note how plausible this is; it says that if you average $f$ over $[q-a,\,q+a]$, as $a\to0^+$ the mean becomes $f(q)$. By L'Hôpital,$$\lim_{a\to0^+}\frac{\int_{q-a}^{q+a}f(x)dx}{2a}=\lim_{a\to0^+}\frac{f(q+a)+f(q-a)}{2}=f(q).$$