Standard calculus textbooks begin by introducing limits, including

  • limits of a fraction as the numerator and denominator approach $0,$
  • limits of a fraction as the numerator and denominator approach $\infty,$
  • limits of continuous functions as their argument approaches a point in their domain,
  • limits of piecewise defined functions,
  • limits at $\infty$ of things like the arctangent function,
  • things like $\lim\limits_{x\,\to\,-\infty} 2^x,$
  • infinite limits,
  • limits of functions whose very artificial-looking graphs are given,
  • limits of a difference as both terms approach $\infty,$
  • limits of areas or lengths as geometric figures approach specified shapes,
  • et cetera.

On the final exam, students are asked to find something like $\displaystyle \lim_{x\to1} \frac{x^2-1}{x-1}$ and reply that it's undefined because it's zero over zero. If that were right, then derivatives would not exist. They miss the central point, which is that in differential calculus limits are there primarily to deal with limits cases where the numerator and denominator approach zero, because that's what derivatives are.

For students who are not there for the purpose of becoming mathematicians, comprehensively covering all topics involving limits that they are able to understand is a mistake: It distracts them from the main event. Are there textbooks written in a way that is consistent with that fact?

(This could applied to many other areas of mathematics as well.)

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    $\begingroup$ One approach is to begin with continuity and use it to communicate from the outset which limits are "easy". Before that explain why 0/0 is interesting (day one of the course). They're not just for the sake of annoying students... in short, when you audience has a short attention span, speak fast. I don't know which text echos my sentiment. Surely there exist open source texts which could be reformatted to force your viewpoint. $\endgroup$ Commented Jul 13, 2017 at 2:29
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    $\begingroup$ If you think that something is central then you have to emphasize that to the students. If something strikes you as less important then you can either skip it entirely or cover it lightly. If they miss a central point despite repeated, explicit attempts on your part to communicate the centrality of that point, it is doubtful that a different text would make much difference. For one thing, students who persistently miss the point are unlikely to actually read the text. $\endgroup$ Commented Jul 13, 2017 at 14:46
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    $\begingroup$ @MichaelHardy Perhaps, though I haven't ever seen a calculus text which didn't concentrate its attention on limits to indeterminate forms on the one hand (which you are discussing) and limits involving infinity on the other hand (a topic which does need to be covered). Piecewise defined tends to make more of a cameo appearance. By the time they get to a final exam they should have seen dozens of 0/0 cases. If they haven't noticed that, would a different sequence help? $\endgroup$ Commented Jul 13, 2017 at 19:28
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    $\begingroup$ I meant "0/0 indeterminate forms", which is what you were referring to. That is the major focus in most treatments of limits in calculus texts (even if the phrase "indeterminate form" tends to be introduced later in the text). $\endgroup$ Commented Jul 13, 2017 at 19:37
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    $\begingroup$ @MichaelHardy It depends on what you mean by "major focus". Slightly less than half of the examples of limits in my version of Stewart are of the 0/0 type, more than any other single type of limit. If you throw in the various limit derivations of derivatives, and the homework you assign emphasizes it, then the majority of limits that the student actually works out in Calc I will be of the 0/0 form. $\endgroup$ Commented Jul 13, 2017 at 19:58

4 Answers 4


I recommend a newer Calculus textbook, which was published earlier this year: DJ Velleman's text, "Calculus: A Rigorous Approach" (link; also mentioned in my answer to MESE 11971). In the introduction, there is a "note to the instructor" as follows:

enter image description here

One of the "novel approaches" in this text is its presentation and notation of and around limits:

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I would say that from the perspective of this text, the "main point" as concerns limits fits into a broader focus around which the topic of Calculus is structured: that which Velleman calls the paradox of precision through approximation. Here is one final excerpt in that direction, which is followed in the text by concrete, motivating examples (and section 2.2 is entitled What Does "Limit" Mean?).

enter image description here

I believe that the subtle but important change(s) in notation, along with the structuring of this text around broader themes in Calculus, make it a worthy candidate for the sort of treatment of limits (and beyond) about which you ask.


I think pedagogically sometimes approaching something with a lot of examples actually makes it easier and gets you there faster than an approach geared only to the final insight (or to a point of generality). Bleecker expresses this well (with data from his teaching experience) in the preface to his book on partial differential equations. He discusses how it is better not to get into Sturm Louisville immediately but approach it indirectly via several examples or smaller problems than the general case. I suspect the same to be true about limits which are really a kind of new breathtaking concept when the student hits them at first.

And I understand your point about the students not knowing the answer to your question at the end of the year. But to be fair one would want to look at an alternate approach and if it does any better. Or what is lost if you just concentrate on that one question. Not even making a statement on what the result would be, but just don't be sure it would go your way.

Finally I don't think missing that question is the most important learning for a calculus student. One can forget some of the theoretical details on calculus and still do more important results (for the mass of the population taking calculus) like finding minima and maxima to curves--key concept and practice in engineering, physics, chemistry, biology and business.

P.s. I struggle with questions like this that have a huge predicate (relevant to pedagogy and worthy of discussion) and then a simple final request. Will the Q and A police come down on me because I discussed the predicate, not gave a specific textbook answer to the question at the very end.


Perhaps you can restructure the final exam question as follows:

It is the case that $\displaystyle{\lim_{x\to1}\frac{x^2-1}{x-1}=2}$. Establish this fact three ways:

  • by simplifying $\displaystyle{\frac{x^2-1}{x-1}}$ algebraically, and using limit properties

  • by identifying the limit as the difference quotient for a derivative $f'(a)$. Be sure to clearly identify $f(x)$ and $a$.

  • by using L'Hospital's rule (perhaps omit this case).

In this way you give the student enough clues to orient their thinking on the exam and show they can connect the limit with various stages of the course.

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    $\begingroup$ Of course, this assumes that the instructor has authority over the final exam, which may or may not be the case. $\endgroup$ Commented Jul 15, 2017 at 20:35
  • $\begingroup$ It might be even better to follow this question up with a more standard "evaluate the limit" kind of question, if for no other reason than to see if students have actually learned anything from the previous question. Of course, you could also present these questions during the course... $\endgroup$
    – Will R
    Commented Nov 1, 2017 at 13:25

It is well known that Abraham Robinson's framework occurs in standard mathematics (ZFC, etc) and Keisler's famous textbook based on rigorous infinitesimals is well aware of the problem you mentioned. Therefore it starts with derivatives before continuity and continuity before limits. Then limits can be defined as a paraphrase of the shadow. In my experience teaching over 400 undergraduates over the past 4 years using this approach, students don't make that kind of mistake when they learn true infinitesimal calculus using this standard textbook.

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    $\begingroup$ "Paraphrasing" is not remotely similar to what I had in mind. I meant covering all technical aspects of a subject at the expense of emphasizing that one aspect is crucially important and the others are not. I don't see how that could be mistake for paraphrasing. $\endgroup$ Commented Jul 13, 2017 at 17:36
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    $\begingroup$ @Michael, the general concern here seems to be "not seeing the forest for the trees". I thought this could be captured more politely using the terminology of paraphrases, but if you feel your formulation is clearer just disregard my suggestion. $\endgroup$ Commented Jul 14, 2017 at 6:59
  • $\begingroup$ My point is not that "paraphrase" is less clear; it is that "paraphrase" has nothing to do with what I was saying. $\endgroup$ Commented Jul 15, 2017 at 16:23

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