So, I realize this can be a broad question, so I'll narrow it down. I have lived in Spain and own several Math textbooks from that country (the equivalent of 8th grade and high school Math). Has anyone here been able to note major differences in the way Limits are taught as well as Introductory Calculus in general? I have noticed that in Spain, they always tend to begin with Sequences before introducing the Limit. I have not seen that approach here in the US. Am I wrong? If so, what book would be closest to the way Calculus/High School level maths are taught in Spain? Also, my impression is (and again, I could be wrong) but just comparing textbooks, the Math level seems to be much more demanding/rigorous in Spain than it is here in the US.
IME, it's a generational thing. When I was in HS in the US in the early 80's, we studied convergent sequences and Cauchy sequences in the year before AP Calculus. (We didn't get to the punchline that the real numbers were the completion of the rationals, so it was not clear at the time what Cauchy had to do with anything.)
Nowadays in the US, it seems like the goal is to expose a broad range of majors to the concepts of calculus at the expense of rigor. Limits are arguably the biggest thing that got pitched. In our defense, recognizing continuity is highly intuitive and then you can retroactively describe the limit as the behavior of a function "near" a member of its domain instead of at the value (which can be gathered by reading a graph or building a table of values). Students majoring in math or engineering then take Real Analysis later in their studies which goes into all of the pedantic details.
In my experience, we don't really expect students to have a strong understanding of limits in Calc 1 in the US. I've always been told to skip the epsilon-delta chapter, and sequences are never brought up until Calc 2.
When I took a real analysis class, we covered limits of sequences first. It seems like a good approach if you want to get a strong understanding for limits.
Lax & Terrell's Calculus with Applications (https://link.springer.com/book/10.1007%2F978-1-4614-7946-8) Starts with:
Chapter 1: Numbers and Limits
- 1.1: Inequalities
- 1.2: The Least Upper Bound Theorem
- 1.3: Sequences and their Limits
- 1.4: The Number e