TITLE: How is the concept of the limit the foundation of calculus?
QUESTION [10 upvotes]: My casual study of mathematics and calculus introduced me to the notion that calculus didn't find a firm foundation until Cauchy, Weierstrauss (et al) developed set theory some ~100 years after Newton and Leibniz.
The concept of limits (and their epsilon-delta proofs) was what allowed calculus to get past the shaky logic of infinitesimals.
(The above is just me laying out what I believe to be true, but please feel free to take issue with any/all of it.)
HERE IS MY ACTUAL QUESTION (more or less).
How exactly do limits save calculus?
When thinking about the derivative, we want to contemplate the rate of change of a function wrt its input. Instead of using an infinitesimal change in the input, we take the limit as the change in the input goes to zero.
When we say "take the limit" we mean finding a band around the function value (+/- "epsilon") and derive a band around the input value (+/- "delta") such that any value of the input beyond a certain point is guaranteed to generate a function value within our desired band (i.e., +/- epsilon). This is all rather loose, but hopefully you catch the drift.
BUT HOW DOES THIS HELP SAVE CALCULUS?
Is it because by saying we can take any ARBITRARY epsilon, we are in effect, saying we can take EVERY epsilon > 0? We exhaust every epsilon all the way down to (but not including) 0. This action reminds me a lot of the idea that 0.999... = 1. 
Sorry if this isn't clear, but if I had to put this in simplest terms, I guess it boils down to: is "arbitrarily many" the same as "infinitely many"?

REPLY [4 votes]: The OP asked (1) how the limits save the calculus, and more specifically (2) how the epsilon-delta definitions save the calculus. The distinction is well-taken and is helpful in clarifying the issues. 
It is well known that there are two main tracks for the development of the calculus: (A) the approach exploiting an Archimedean continuum, where for example limits and continuity are defined via epsilon-delta; and (B) the approach exploiting what could be called a Bernoullian continuum, i.e., a continuum featuring infinitesimals. Johann Bernoulli was the first one to use systematically infinitesimals and differentials to found the calculus, whereas his teacher Leibniz exploited both (A-track) exhaustion techniques and (B-track) infinitesimal techniques. 
The distinction between the OP's questions (1) and (2) is significant because limits are actually present in both the A-track and the B-track. Thus, one can define the limit of a function $f(x)$ as $x\to 0$ via epsilon, delta, but one can also define it as the standard part of $f(\alpha)$ where $\alpha\not=0$ is an infinitesimal. 
This is a point that is often overlooked in historical commentary as well as in some of the answers above. Thus, to Cauchy the primitive concept was that of a variable quantity, and he defined both limits and infinitesimals in terms of this primitive concept. Cauchy's definition of continuity of $f$ is not at all what one would expect it to be based on traditional historiography. Cauchy says that $f$ is continuous if every infinitesimal increment $\alpha$ always produces an infinitesimal change $f(x+\alpha)-f(x)$ in the function. 
To answer the OP's question, the reason epsilon-delta definitions so to speak "save" the calculus is because they are formulated in an A-track fashion. This "saves" the calculus because mathematicians around 1870 have not yet developed a theory of infinitesimals that was thought satisfactory. Therefore some of them thought that the A-track was the only one available. As mentioned in some of the other answers, this foundational difficulty was overcome about 60 years ago by Abraham Robinson, so today it is the infinitesimal B-track that "saves" the calculus as far as both the students and the professors are concerned.