Query about the electric field of a uniformly charged rod at the ends of the rod

I found the electric field of a uniformly charged thin rod to be, $$E=\frac{Q}{4\pi \epsilon_0 (a^2-x^2)}$$. Where $$L_{rod}=2a$$ and $$x$$ is the distance away from the centre of the rod, along the its axis (the horizontal or x axis in this case). My question is, according to this model the electric field at one of the ends of the rod is infinite for even a small charge. Is this really the case or do you say that this model is only valid for $$x>a$$ and at $$x=a$$ you can take all the charge to be at the centre or at $$x=0$$. However, I dont believe we can make this assumption as it should still hold for $$x>a$$ and if it did the electric field for a thin rod would be the same as a sphere.

So my question is, is the electric field at one of the ends really infinite, or is there an upper bound and if so, how do you calculate it?

Note that the rod is "infinitely thin", which is the source of trouble. Normally rods are a bit thick. The formula should be correct anywhere along the axis, and you get approximately spherical field for $$x>>a$$.