It basically means whatever it's supposed to in the given context. If you look at it, you realize that there are no words in that proof. It's just the x-y plane minus the part that you can draw. On taking derivatives Because we pretend we are Newton doing calculus. Limits don't exist.
Limits are a fiction invented by analysts.
Derivatives are purely symbol manipulation on polynomials. That's how thousands of first semester students get through calculus. Yes, I know this is an algebra class, but let's differentiate anyway. Epsilon is the number that's so small that its square is zero even though it's not zero. You can do most of calculus that way.