>>11834683Don't start with Analysis books, they're for people who already know Calc and some math in general.
As for your question, yes they introduce limits. The difference is rigour and the level of the explanation. In introductory courses (and in books like Stewart) limits are defined as what the function approaches as the value considered approaches some other value. Then you're given the rules to work with them algebraically, same with differentiation and such.
In more rigourous books (Apostol, Spivak) you get this explanation but then you rigourously define what "approach" means, you derive the rules for algebraic operations, etc. Basically as the concepcts are introduced. (In stewart, for example, the rigorous definition of a limit is relegated to an appendix).
The problem sets in the books reflect this. Books like Stewart are more computational in nature to drill the patterns into you. The other books have some of this as well, but they also ask you to prove various statements, give you some more complicated problems, etc.
>>11836677Spivak is absolutely not between the two, you "gathered" garbage. Apostol is rigurous and reads like a math textbook with theorem -> proof, etc but has a fair bit of applications and ties to the real world. Spivak is just straight pure math for the most part. It's a fun book though.
Also, engineers at my uni use Apostol in their first calc class
