Let's have function $$ \tag 1 f(x, t) = \frac{1}{\sqrt{i\pi t}}e^{i\frac{x}{t}}. $$ How to prove that $(1)$ is delta-function when $t \to 0$?
I have tried to use "usual" properties of delta-function, $$ \int f(x, t)dx = 1, \quad f(0, t \to 0) = \infty , \quad f(x \neq 0, t \to 0) = 0, $$ but there is the problem with the third identity: the function strongly oscillates instead of be zero. But peolpe say that $(1)$ becomes delta-function when $t$ goes to zero. So how to show that?
Addition.
Maybe it's good to make Fourier transformation of $f(x, t)$ and then to get that $\lim_{t \to 0^{+}} f(p, t) = 1$, which is the Fourier image of delta-function.