The definition of limit: "lim x -> a f(x) = c " means for all epsilon > 0, there exists delta > 0 such that for all x, if 0 < |x-a|<delta then |f(x) - c| < epsilon.

The definition of derivative: f'(x) = lim h -> 0 (f(x+h) - f(x))/h

That is, for all epsilon > 0, there exists delta > 0 such that for all h, if 0 < |h| < delta then |(f(x+h) - f(x))/h - f'(x)| < epsilon.

At no point do we divide by 0. h never takes on the value 0.

You said " Dividing by zero doesn't produce a contradiction"

Several of these links include examples of contradictions. There is no authority required.

For example:

A Contradiction. Suppose we define 1/0 = q

for some real number . Multiplying on both sides of the equation gives 1 = 0 * q = 0

which is a contradiction (to 1 and 0 being different numbers).

Since you asked this of papermachine, it seems reasonable to reflect it back:

Are you asserting this as somebody with strong knowledge of mathematics?

What are you expecting me to update on?

Sigh. Consider this my last reply.

• mstevens' links have several demonstrations that division by zero leads to contradictions in arithmetic.
• my link (singular) demonstrates that the definition of a derivative never requires division by zero.
• Qiaochu's proof in a sibling thread that the only ring in which zero has an inverse is the zero ring.

So what exactly is our disageement?

That you continue to say things like "A derivative -is- a division by zero" and "division by zero can be valid", as if they were facts. Yes, you may have been taught these things, but that does not make them literally true, as many people have tried to explain to you.

Incidentally, even if I were certain exactly what you're trying to convince me of and it was something I didn't already agree with, your links are nothing but appeals to authority, and they wouldn't convince me -anyways-. They lack any kind of proof; they're just assertions.

Whose authority am I appealing to in my (singular) link? Doctor Rick? I imagine he's no more a doctor than Dr. Laura. (I actually knew one of the "doctors" on the math forum once, and he wasn't a Ph. D. (or even a grad student) either; just a reasonably intelligent person who understood mathematics properly.) The only thing he asserts is the classical definition of a derivative.

Or maybe you were just giving a fully general counterargument, without reading the link.

EDIT: It's simply logically rude to ask for my credentials, and then treat every single argument you've been presented as an argument from authority, using that as a basis for dismissing them out of hand.

Limits and calculus isn't what I think of, at all, when I think of division. I pretty much limit it exclusive to the multiplicative inverse in mathematical systems where addition and multiplication work like you think they ought to. There are axioms that encompass all of "works like you think they ought to", and a necessary one of them is the multiplicative inverse of zero is not a number.