My friend and I were studying for our EM test when we started to think about what happens to the electric field near an infinite line of charge.

$$E = \frac{\lambda}{2\pi\rho\epsilon\_{0}}$$

As you get close to the line of charge, it seems like the electric field strength goes to infinity.

We were wondering if this means that, even in the classical picture of physics, two electrons could never touch because it would require an infinite amount of energy. Is this correct?