# Calculus limits taught in the US vs Spain?

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.

• The level of math taught in high school in Spain is generally higher than what is usually taught in the US, although the US is far more heterogeneous. In Spain both basic linear algebra and calculus are examined on the exams used to determine entrance in the university. However in the standard curricula in the Community of Madrid (which is fairly typical) limits are taught in high school but sequences are not, so I'd question the premise that sequences are generally taught before limits are. Commented May 8, 2021 at 11:14

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.

• This is by far my preferred approach. To me, it seems that teaching limits first is like forcing people to learn how to construct the reals before allowing them to multiply decimals, or perhaps a closer analogy would be constructing the reals before using pi. Commented May 4, 2021 at 14:13

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
• "I've always been told to skip the epsilon-delta chapter..." That's sad. I suppose you don't have any students who are math majors, or any students who transfer to schools with less watered-down standards.
– user507
Commented May 2, 2021 at 13:39
• This includes when I taught Calc 1 at Cornell! Students who were interested in becoming math majors had the option of taking Honors Calculus (out of Lax-Terrell). Covering epsilon-delta properly takes time that some might argue is better spent elsewhere. Commented May 2, 2021 at 14:53
• @BenCrowell In many (possibly most) universities in the US students do not declare their major until after their first (or in some cases second) year of studies. In such a setting, almost by definition nobody taking 1st-year Calculus is a math major, at least not yet. Commented May 5, 2021 at 20:17
• ...@BenCrowell for example, the University of Michigan, UC Berkeley, and Yale are three well-known and highly-respected universities in which students typically declare their major at the end of the sophomore year. So "math majors in Calc 1" is essentially a non-existent category at all three of those institutions. Commented May 5, 2021 at 23:43
• ...Now you might be thinking, okay, but there are some students who intend to be math majors, even as incoming 1st-year students, and surely they should be seeing an epsilon-delta treatment, right? Sure, but most of those students are not taking Calc 1. They usually either (a) place out of it or (b) take an honors-level Calculus course, in which they do encounter rigorous definitions and some proof. Commented May 5, 2021 at 23:45