I know that when I learned calculus the hardest part was reconciling the notion of infinitesimals with algebra. The delta/epsilon definition of a limit was unsettling. It's a shame. Much of calculus is easy to grasp intuitively - Riemann integration is easy to see and explain.
Yes, the concepts of limits and derivatives are simple and intuitive, but the concept of the epsilon/delta proof just completely soars over students heads. I didn't learn the epsilon/delta stuff until after all that other stuff, and I don't feel like it was the wrong order of things.
That and the prof I had never just really explained that epsilon just means "really insanely small" which probably would've helped.
I had to learn the epsilon-delta definition before I really understood limits. As far as insanely small goes, as you must know, the definition is equivalent to "for every epsilon < M, there exists a delta >0, ..." where M is some positive constant (it could be 1 or 1/10^100...). Sometimes it helps to play with the quantifiers...
In my university, the first calculus course for most of the people uses only an intuitive definition of limit (without epsilon/delta or epsilon/n_0). Also the definition of derivative and integral is intuitive. The idea is that the students can do the calculations following some rules. And then see some applications, like the tangent line of a function or the area below a function.
The first calculus course for engineers and exact science students is more difficult, but the discussion of the technical details of the limit is very small. Perhaps calculate the limit by definition only for 2 or 3 easy functions, and then just use the algebraic properties of the limit.
The second course for Math and Physics students has more time assigned to the calculation of limits by definition.
Actually, limits and infinitesimals are two very different explanations/foundations for calculus. Infinitesimals are much more intuitive.
Up to last decade, limits were the only rigorous enough explanation. Until someone invented an equally rigorous reconstruction of calculus with infinitesimals.
