One difficulty students had that I encountered as a TA was some garbled prerequisites. All of the epsilon-delta definitions are written in propositional logic,

One difficulty students had that I encountered as a TA was some garbled prerequisites. All of the epsilon-delta definitions are written in propositional logic, and are often '2nd order statements', (that is they have nested quantifiers). This is an entirely new formal language, and its usage is very different than english. It needs to be carefully explained, but standard texts like Stewart just dress it up to look kind of like english and carry on. In fact the mathematics curriculum DOES acknowledge that you have to teach most students this if you want them to understand it: at my university it lived in the discrete math course, which used Rosen. He devotes an entire chapter on propositional logic, and spends literally 60 pages gradually building up the complexity to arbitrary nested quantifiers; the definition of the limit appears at the end of this. Unfortunately, discrete math also makes heavy use of sequences and series, so that Calc 2 is a prerequisite of the course... thus my program's student-victims would spend a year taking calculus and not understanding much of the formalism before they were even allowed to take the course that explained the language the basic definitions of calculus were written in! Ugh. I think Stewart and Rosen are pretty mainstream textbooks, so i suspect this problem is very common. Perhaps you could point it out at your next faculty meeting and shuffle some prerequisites around; we'll start a math-revolution! :)