Question on a modified definition of divergence. I've been looking at this question for a few days but I still can't fully understand it. I know that divergence is defined as flux per unit volume, which corresponds to the same limit you see below but multiplied by $3/(4\pi e^3)$, which is 1 divided by the volume of the sphere or radius e. So what does it represent when we use area instead of volume? I really have no clue about how to approach this question.

Let $\bf{F}$ be a smooth 3D vector field and let $S_{\epsilon}$ denote the sphere of radius $\epsilon$ centred at the origin. What is
$$ \lim_{\epsilon \to 0+} \frac{1}{4 \pi \epsilon^2} \int \int_{S_{\epsilon}} \bf{F} \cdot d \bf{S}$$
Note here we are taking the limiting value of the flux per unit surface area, not the flux per unit enclosed volume. You must explain your reasoning.

Thank you very much for any help you may give me!

It all depends on the definition. In my cursus divegence is differential operator $$\mathrm{div} \,\mathbf F=\nabla\cdot \mathbf F.$$  Then we can via Gauss theorem make a connection of flux through surface with integral of divergence in the volume bounded by that surface (mathematically, we should talk about manifolds with boundaries, but this another topic). By studying the integral you posted you can obtain that the limit is indeed $ \mathrm{div} \,\mathbf F$.
