"That change c never truly goes away, but it is so small that it doesn't really matter: if you make c small enough, the derivative is basically 8. So we're just going to pretend that all the cs are so small they have become zero."

My first calculus teacher taught us derivatives in a similar way - but I have to say that, for me, this imprecise language confused me. When and why is it OK to pretend that c is zero!?

Further on, the official definition using a limit claims that the "c->0" means "make c as small as possible". But that's not what it means. Again, this is imprecise language that confuses more than it explains. What does "as small as possible" even mean? If we want to make it as small as possible, why not set it to 0!? How small is small enough?

I think somewhere in this article there should be a precise definition of what a limit is, using epsilon and delta.

It should be saying something using the words "arbitrarily close" and "sufficiently small".

Something like : "we can make the approximation on the left arbitrarily close to the expression on the right by choosing any value of c sufficiently close to 0, even though the expression might be undefined when c=0. Epsilon: you tell me how close you want the approximation to be. Delta: I tell you how small c needs to be".

There used to be a great website that was precise but also informal: karlscalculus.org - but sadly it appears to be down now.