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.

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. [...] 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. [...] 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.