The following is from Chapter 16.4: Green's Theorem in the Plane, Thomas's Calculus, 14th Edition:
Circulation rate around rectangle $\approx \left( \dfrac{\partial{N}}{\partial{x}} - \dfrac{\partial{M}}{\partial{y}} \right) \Delta x \Delta y $.
We now divide by $\Delta x \Delta y$ to estimate the circulation rate per unit area or circulation density for the rectangle:
$\dfrac{\text{circulation around rectangle}}{\text{rectangle area}} \approx \left( \dfrac{\partial{N}}{\partial{x}} - \dfrac{\partial{M}}{\partial{y}} \right)$
We let $\Delta x$ and $\Delta y$ approach zero to define the circulation density of $\mathbf{F}$ at the point $(x, y)$.
The above explanation is in reference to the following image:
And here is some further context:
What I don't understand is why we let $\Delta x$ and $\Delta y$ approach zero to define the circulation density of $\mathbf{F}$ at the point $(x, y)$? If we divide the circulation rate around the rectangle $\left( \left( \dfrac{\partial{N}}{\partial{x}} - \dfrac{\partial{M}}{\partial{y}} \right) \Delta x \Delta y \right)$ by the rectangle area $\left( \Delta x \Delta y \right)$, then the $\Delta x \Delta y$ terms just cancels out, so that $\dfrac{\text{circulation around rectangle}}{\text{rectangle area}} \approx \dfrac{ \left( \dfrac{\partial{N}}{\partial{x}} - \dfrac{\partial{M}}{\partial{y}} \right) \Delta x \Delta y }{ \Delta x \Delta y } = \left( \dfrac{\partial{N}}{\partial{x}} - \dfrac{\partial{M}}{\partial{y}} \right)$.
So I don't understand why the authors throw in "We let $\Delta x$ and $\Delta y$ approach zero to define the circulation density of $\mathbf{F}$ at the point $(x, y)$." From reading this sentence, it seems that they're trying to do something similar to the limit definition of derivatives, but, as I said, I'm not sure what this is supposed to mean in this context, nor am I sure of what purpose it's supposed to serve since, as I just demonstrated, the algebra neatly gets us the circulation density without limits?
I would greatly appreciate it if people could please take the time to clarify this.