In the limit $ r \to 0$, the energy is indeed infinite. But the idea is that, you cannot have $ r \to 0$ in any 'classical' calculation. Note that when we talk of quantum mechanics, no charge can be point-like, because it will violate uncertainty principle. So, in this case, the resolution is that there is a quantity, known as "classical electron radius" given by $ r_{0} = \frac{e^2}{m_{e}c^2}$, it has the numerical value $2.82 \times 10^{−13}$ cm, about one one-hundred-thousandth of the diameter of an atom. So, one still has a finite value of U.

I would highly recommend reading Chapter 28 Volume II of Feynman Lectures to understand this. No one can answer this better in my opinion. http://www.feynmanlectures.caltech.edu/II_28.html

In the limit $ r \to 0$, the energy is indeed infinite. But the idea is that, you cannot have $ r \to 0$ in any 'classical' calculation. Note that when we talk of quantum mechanics, no charge can be point-like, because it will violate uncertainty principle. So, in this case, the resolution is that there is a quantity, known as "classical electron radius" given by $ r_{0} = \frac{e^2}{m_{e}c^2}$, it has the numerical value $2.82 \times 10^{−13}$ cm, about one one-hundred-thousandth of the diameter of an atom. So, one still has a finite value of U.

I would highly recommend reading Chapter 28 Volume II of Feynman Lectures to understand this. No one can answer this better than that book in my opinion at this level without invoking any ideas of quantum theory. http://www.feynmanlectures.caltech.edu/II_28.html