Suppose you are given a black box with some unknown signal $f(t)$ in it, and the only thing you can do is to provide another signal $in(t)$ as input, in which case the black box will output the scalar product between the two: $\langle f(t), in(t)\rangle$.
What kind of input signal should you use to get the value of $f(\Delta)$?
The three plots on the top left show the unknown function $f(t)=\cos(3t)$. The next three plots show possible candidates for $in(t)$: rectangle, triangle and sinc functions which can be modified using sliders $a$ and $\Delta$. Notice that the integral of these three functions is always 1, whatever $a$.
In the three bottom left plots, we multiply our three functions $in(t)$ with $f(t)$. Then we compute and print the scalar product as the integral of this product, i.e. the signed area in blue.
When $a$ grows, we see that our three functions, although not fully identical, tend to have the same effect when used in the integral: only their values very close to their maximum contribute to the result.
When $a$ tends to infinity (here, to 20), these functions can no longer be plotted. They are therefore termed as a Dirac impulse $\delta(t)$ and symbolically represented on the right plots as an arrow, the amplitude of which is set to the integral of the functions: 1 on the top plot and $f(\Delta)$ on the bottom plot.
As a matter of fact, we see that when $\Delta$ is set to 0, all integrals tend to $f(0)$:
When $\Delta$ is modified, all integrals tend to $f(\Delta)$:
The ideal input function $in(t)$ is therefore $\delta(t-\Delta)$.