electromagnetism - Derivation of the Eikonal Equation - Neglecting Derivative of $epsilon$?

In "Principles of Optics" by Max Born and Emil Wolf, the eikonal equation is derived from Maxwell's equations. In there, the authors use the approximation of low wavelengths / high wavenumbers, to neglect some terms in the wave equation and arrive at the Eikonal equation. One of those neglected terms is the term $\nabla \log \epsilon$, and the reason for that is that it is smaller than the "wavenumber" $k$ of the solution. We then proceed to derive the Eikonal equation and fermat's principle and it is happily ever after used in geometric optics.

I may show some steps in the derivation: In general linear inhomogenous media, the wave equations that follow from Maxwell's equations are given as: $$ \Delta \vec{E} - \frac{\epsilon \mu}{c^2} \ddot{\vec{E}} + (\nabla \log \mu ) \times (\nabla \times \vec{E}) + \nabla (\vec{E} \nabla \log \epsilon) = 0 $$ Plugging in $\vec{E} = \vec{E}_0(\vec{r})e^{i(k\chi(\vec{r} - \omega t)}$, looking only at the imaginary part of the equation and neglecting everything that isn't of order $k^2$, you arrive at the Eikonal equation. In a similar way (see my other question on that), you make an approximation to arrive at the commonly used "approximated wave equation" (I will for now call it like that) in a medium:

$$ \Delta \vec{E} - \frac{\epsilon (\vec{x}) \mu (\vec{x})}{c^2} \ddot{\vec{E}} = 0 $$

So here is my question: Think of the case of a sharp material change in surface (that would be a discontinuity of $\epsilon$), and of a plane wave that is refracted at this plane (we all know Snell's law of refraction is applicable here). All in all, the whole solution in this case (incident wave and refracted wave and reflected wave) do have a discontinuity at the plane.

This solution was obtained by solving the wave equation for both materials and then glueing them together, using the right continuity conditions. Similar, one could solve the given general wave equation given at first. Our "approximated wave equation" is not able to describe this solution, it will not yield a discontinuity. So far everything's fine, we made an approximation (neglecting the terms containing derivations of $\epsilon$), we can't expect the equation to still be as powerful as it has been before.

But then (and that really buffles me, and is the core of my question): Making stronger assumptions and neglecting even more terms, we arive at a theory (geometrical optics) that can describe refraction at a surface perfectly well.

Why is it that geometrical optics can describe something like refraction better than the "approximated wave equation", when geometrical optics are even a bigger approximation of said wave equations? And why do geometrical optics describe surfaces of edges of different media, when in there you have discontinuities in the refractive index, and thus can't make those said approximations?