To my question:
There are many functions that represent the Dirac delta "function", like the square wave function:
$$\delta_n(t)=n, -\frac{1}{2n}<t<\frac{1}{2n}$$
And clearly for any value of $n$: $$\int_{-\infty}^{\infty}dt\delta_n(t)=1$$
and in the limit as $n \rightarrow \infty$, $\delta_n(t)=0, \forall t $ except $t=0$.
So how would i prove this? I do have something in mind that would be simply to evaluate the integral with $n \to \infty$ but I'm not quite sure, so just wanted to ask if this would be correct, and the following all could be proved similarly?
Like the Resonance: $$\delta_n(t)=\frac{n/\pi}{1+n^2t^2}$$ or the Sinc Squared: $$\delta_n(t)=\frac{sin^2(nt}{n\pi t^2}$$ or the Gaussian: $$\delta_n(T)=\frac{n}{\sqrt{\pi}}e^{-{n^2t^2}}$$
But the following example couldn't be evaluated that way and so i was just wondering if there is a general method? And How would you show that this function is a representation of the Dirac delta function?
$$\frac{e^-\frac{x^2}{\epsilon^2}}{\sqrt{\pi}\epsilon}, \epsilon > 0$$