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If you find the limit is 2 for a given function, wouldn't this be the same as $2 + \epsilon$ with $\epsilon$ being a negligible value? This different way of defining limit-like behavior seems rigorous enough, but it took until Abraham Robinson in the 60s to really define the foundation for nonstandard analysis. My question is: what really is the difference?

rb612
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    The difference is that I could explain to you exactly what "limit" means, but "negligible value" doesn't mean anything. – Jack M Jun 17 '16 at 20:34
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    I imagine the problem arises when you have a "negligible value" divided by a "negligible value" yielding a real number i.e. a derivative, or adding an infinite number of "negligible values" together to again yield a real number i.e. an integral. This needs to be well defined, so we need rigorous definitions. – JasonM Jun 17 '16 at 20:38
  • @JackM thanks for your answer, but how exactly would a limit be different from a negligible value? I get epsilon-delta and that is rigorously defined, but we're essentially taking the standard part function of the infinitesimal to obtain the exact value of the limit, from my understanding. And thank you Jason, that does make sense. I just don't really understand how limits are very much different. The error terms become infinitesimal in an infinite Riemann sum, but we don't treat them as infinitesimal, we just say they are zero. That is the only difference I see. – rb612 Jun 17 '16 at 20:42
  • The difference is the machinery to make the intuitive notion of an infinitesimal rigorous is more complicated,at least in the eyes of some, than the machinery of standard analysis. – Conrad Turner Jun 17 '16 at 20:43
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    The difference is that $\varepsilon,\delta$ are real numbers, but infinitesimals are not real numbers with the exception of $0$. Then you need a new axiomatic set theory for this, to handle hyperreal numbers properly. This is a lot more than work and sophistication to something that can be explained easily just using real numbers. – Masacroso Jun 17 '16 at 20:59
  • Even for mere existence, we need (a weak form of) the Axiom of Choice. – André Nicolas Jun 17 '16 at 21:39
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    @AndréNicolas, you need the axiom of choice even for such an innocuous fact that the Lebesgue measure is countably additive. Sometimes we see a double standard at work in applying the AC criticism. – Mikhail Katz Jun 19 '16 at 08:32
  • If $2$ is the limit, then $2 + \epsilon$ is not the limit; $2 + \epsilon$ is only infinitesimally close to the limit. –  Jul 16 '16 at 07:59
  • @JackM In any field outside pure math 'negligible value' means irrelevant or immeasurable. In math it just means arbitrarily small. –  Nov 26 '16 at 21:07
  • https://math.stackexchange.com/questions/661999/are-infinitesimals-dangerous –  Apr 29 '17 at 05:51

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There were various theories extending the real numbers to include infinitesimals throughout the period from 1870 until 1960, but Robinson was the first to introduce a system that can be used in analysis. Earlier systems were critized by Klein, Fraenkel, and others on the grounds that they were not proven to satisfy the mean value theorem for, e.g., infinitesimal intervals. Robinson's framework satisfies this and more.

Mikhail Katz
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