Dirac Delta function in the form of rectangular function

Question

this is the image

The dirac function is an impulse and defined on a point. i am not able to understand the equivalence of the functions in the image . Explaining why they are equal would be helpful. Thanks.

Answer 1

Intuitively, the Dirac delta pulse is a thin (width zero) and high (height infinity) pulse with integral equal to one.

$\Pi$ is a not so thin (width one) and not so high (height one) pulse with integral equal to one. When you modify it as $\frac{1}{\epsilon} \Pi(\frac{x}{\epsilon})$ you get a thinner and higher pulse, with width $\epsilon$ and height $\frac{1}{\epsilon}$ and still integral equal to one.

As $\epsilon \to 0$ this pulse gets even thinner and higher, and in this way gets closer to being a Dirac pulse.