From what I understand after thinking about this, delta epsilon really seems to formalize the notion of an infinitesimal. The constraint $0<|\delta-c|$ combined with the fact that there is no real number that doesn't satisfy the delta-epsilon definition, then don't these two combined facts mean that an infinitesimal exists just not within the reals? In other words, there is a value that is greater than zero that is smaller than all possible real numbers (from my understanding), and the only value this could be is an infinitesimal. But I know that infinitesimals don't exist within the realm of real analysis. So where is my thinking wrong?
Answer
The opposition "limit versus infinitesimal" is a bit of a false opposition because limits are present both in the A-track approach working with an Archimedean continuum (the real numbers), and in the B-track approach working with a Bernoullian continuum (i.e., an infinitesimal-enriched one).
In the A-track, limit is defined via epsilon-delta definitions.
In the B-track, limit is defined in a more straightforward way using infinitesimals.
For example, $\lim_{x\to0}f(x)$ can be defined simply as the standard part of $f(\alpha)$ where $\alpha\not=0$ is infinitesimal.
Both epsilon-delta techniques and infinitesimals provide rigorous ways of handling the calculus. The relation between them can be stated as follows. The epsilon-delta techniques and definitions are a paraphrase of infinitesimal techniques and definitions.
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