electrostatics - Confused about the electric field at the surface of a conductor

We know that the electric field at the surface of a conductor only have a normal component equal to $\rho/\varepsilon$ (finite number).

But let’s consider the point $\text{P}$ (at the surface of a conductor). Assume that there is a charge at an infinitesimal distance from the point $\text{p}$. We can obtain the field at the $\text{P}$ by the formula $E=Kq/r$. Obviously, $E\sim1/r$. So the normal component of the field is infinite. Now if we add the field due to other charges, it will remain infinite. So where could I be possibly wrong?