Bendixson's condition for existence of limit cycle for a nonlinear system

Question

I encountered this example in Slotine,Lee:Nonlinear Control book. Consider the nonlinear system $$\dot{x_1} = g(x_2) + 4x_1x_2^2$$ $$\dot{x_2} = h(x_1) + 4x_1^2x_2$$ Is there a limit cycle on phase plane?
The solution calculates $\frac{\partial{f_1}}{\partial{x_1}} + \frac{\partial{f_2}}{\partial{x_2}} = 4(x_1^2 + x_2^2)$ and by Bendixson's theorem suggests than this value is strictly positive (except at the origin), so the system has no limit cycle.

Here is my question: What about the origin? Why origin in this case is not considered as a point which makes the application of the theorem useless? The origin is not even an equilibrium for the system. If it was, what difference would it make?

Excerpt from the book:

Answer 1

The fact that the origin might be or not an equilibrium is irrelevant for the criterion in terms of the divergence.