How is the concept of the limit the foundation of calculus? 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"?

Another way to avoid limits is to consider dual numbers, where there is a number $\epsilon \neq 0$ with a vanishing square $\epsilon^2 = 0$. This gives an algebraic model of calculus because one may derive Taylor's theorem and the derivatives of elementary functions by pushing the argument into the dual domain.
