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To give an example of what I mean, I'll answer a similarly worded question: “is there a pre-calculus introduction to the derivative?” I would say yes, since there already are the ideas of a slopes of a line, the tangent and secant lines. The major difference in calculus is that it includes the limit into the equations.

But, what about limits, and specifically the formal definition of a limit? What would prepare students for it? I suppose the absolute value would need to be included since we’re talking about being within values of epsilons and deltas, but what would be a good starting off point?

If you had to teach the formal definition of a limit, how would you end the phrase, “As you know, from precalculus…”

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    $\begingroup$ I barely teach the formal definition of limit in calculus. I think of it as an analysis topic (much more advanced). I think it would be helpful if we played around with infinity more, but no, not this formal definition. $\endgroup$ – Sue VanHattum♦ Sep 2 '19 at 16:34
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    $\begingroup$ I think the primary thing to get across in precalculus is the precise idea of a function. Understanding the concept of a "hole in the graph" and/or our ability to define a function by more than a mere formula are crucial to understand the later conversation about the precise definition of a limit. Limit of what ? First understand the what. $\endgroup$ – James S. Cook Sep 2 '19 at 19:41
  • $\begingroup$ “is there a pre-calculus introduction to the derivative?” I would say yes, since there already are the ideas of a slopes of a line, the tangent and secant lines. Without limits, it's a little tricky to give an actual definition of the tangent line (as opposed to a conceptual description) without introducing limits. One approach is the one used by Marsden in Calculus Unlimited. For a presentation, see my book Brief Calculus, lightandmatter.com/calc , in the appendix labeled "A- Detours." $\endgroup$ – Ben Crowell Sep 2 '19 at 20:19
  • $\begingroup$ This is more of an opinion than an answer, hence I am putting it into a comment: limits really aren't appropriate content for a precalculus class. Limits are where calculus begins (typically), hence they shouldn't be taught in precalc, but should be reserved for the first week of calculus (or thereabouts). $\endgroup$ – Xander Henderson Sep 2 '19 at 20:53
  • $\begingroup$ These are probably a bit more sophisticated than you're dealing with, but others reading this might be interested in the following three books, all published in 1964: Limits. The Concept and Its Role in Mathematics AND Limits and Continuity by William K. Smith AND Functions, Limits, and Continuity by Paulo Ribenboim. All three are reviewed in Mathematics Magazine 38 #5 (November 1965), pp. 311-313. $\endgroup$ – Dave L Renfro Sep 3 '19 at 17:32
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Perhaps what would best prepare a student for the formal $\epsilon{-}\delta$ definition of a limit is an animation which shows both $\epsilon$ and $\delta$ approaching $0$, $\epsilon \neq \delta$, something like this (apologies for the too-high speed):

         
          (Image from here.)
(This would be more effective with a non-linear curve, but I couldn't quickly find such an animation.)

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Precalc courses differ a lot, so it's hard to say something is included or not. After all there was a time when there didn't even exist a precalc course (different from a strong algebra 2, trig, and nalytic geometry sequence). Then again there are some precalc course that spend a reasonable amount of time on theory of functions (and relations), inequalities, and include limits and even simple differentiation, anti-derivatives. Basically an intro to calculus.

The sequence I had was: *Pre-algebra (solving for single variable in first order equations) *Algebra 1 (lotta x-y lines and maybe the quadratic equation) *Geometry (lotta triangle proofs) *Algebra 2 (logarithms, exponents, sequences, basically "college algebra", except a very short intro to vectors (needed to support HS physics *Trig (1 sem) *Functions (1 sem) *Analytic geometry *AP Calculus (AB or BC)

The upper track kids did the sequence above in 7-12. Taking "freshman algebra" as eighth graders and algebra 2 and trig being combined from 3 to 2 semesters in a course designed for that track. The standard track started with algebra in year 9 (prealgebra in 8). Finished with either functions (2nd semester senior year) or with analytic geometry (kids taking algebra 2 trig or maybe doubling up a semester somewhere...I know "functions" was required for AP Chem so that provided a driver for some kids who were not in 8th grade algebra. I believe it was also acceptable to stop with geometry in year 10, for the lowest track.

The "functions" course corresponded to a strong pre-calc course as described above (but not some real analysis monstrosity). And also maybe just gave some time/maturity/practice that would be helpful before going to the shock of college calculus (BC is very comparable to a classic college calc class). Similarly for analytic geometry. I mean in theory, you could go straight from trig to a classic calc course, with analytic geometry in the text also. But I think you'd have a lot of attrition, given the sudden jump in difficulty.

As a single datum, the (strong) local public high school I went to in the 80s, did have limits in precalc. And at a reasonably rigorous level, more than just "derivatives are like tangents". Talking about left and right hand limits and where a limit doesn't exist. And the darned Le Hospital. But without such a concentration that people become more interested in the rigor as opposed to the result).

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  • $\begingroup$ I believe it was also acceptable to stop with geometry in year 10, for the lowest track. --- When I was in high school (in the U.S., 1973-77), these classes were for those considering attending college (maybe 40%-60% of all students, with 15%-30% actually attending college). Probably 30%-40% never took geometry and 20% never took algebra 1. The 2 required math classes for graduation were taken from General Math I, II, III (not sure if there was a IV, but a HS I taught at in 1988-89 had a IV) and 2 years of consumer math (for those in typing and other secretarial-prep classes). $\endgroup$ – Dave L Renfro Sep 3 '19 at 17:43
  • $\begingroup$ You're probably right. It has been a long time for me, so don't recall the details. But there was consumer chem (called cookbook chem) and then there were lower track courses for world history and math also. We had a decent shop department and even had auto shop (sort of a center school for that). $\endgroup$ – guest Sep 4 '19 at 12:16
  • $\begingroup$ Incidentally, I think the state requirement (where I lived) for graduation was increased from 2 years of math to 3 years of math sometime in 1980s, but I'm not sure exactly when. I know it was 3 years where I taught in 1988-89, this in a neighboring state. Also, college admissions were fairly specific about math requirements, with "3 years including geometry" being a common requirement for 4-year colleges and universities. Of course, the more selective colleges/universities either had a higher minimum or it was well known that you had little chance of getting in without at least algebra 2. $\endgroup$ – Dave L Renfro Sep 4 '19 at 12:50
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The closest approach (pun intentional) to a limit in precalculus is when you're dealing with asymptotes:

https://youtu.be/gOQffI21rpc?list=PLKXdxQAT3tCuJku9nTlRZgx_RjGZ7djMc

The advantage here is that it really drives home the idea of the limit of a function as a prediction of what the function value will be.

Limits at finite values are harder. You might be able to talk about rational functions, for example, but you'd have to really get students to understand that $\frac{x^{2} - 4}{x + 2} \neq x - 2$.

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