# How long would it take to teach proper limit calculations?

This question arose from discussion of this question.

How long would it take you to teach typical undergratuate (calculus) students the difference between the following two calculations?

$$\lim_{x\to \infty}\frac{x+2}{x(x-1)}=\frac{\infty}{\infty^2}=\frac{1}{\infty}=0$$ versus $$\lim_{x\to\infty}\frac{x+2}{x(x-1)}=\lim_{x\to\infty}\frac{x^{-1}+2x^{-2}}{1-x^{-1}}=\frac{0-0}{1-0}=0$$

I'm happy to accept different interpretations of 'long' and 'typical' (please give an indication of what they mean to you).

By 'teach' I mean such that the students properly grasp why one solution is better than the other, rather than simply that they remember that 'the teacher told us to do it this way' and can do the calculation.

Justification from research if possible would be nice, but I'd be interested to hear personal experience too.

• I'd expect a fast trip through some other "$\infty/\infty$" calculations giving contradicting nonsense, plus a lecture about "$\infty$ is not a number, as it doesn't behave" should be (mostly) enough... – vonbrand Nov 15 '15 at 19:35
• I prefer the middle-ground: I teach the hierarchy of functions (constants, logs, polys by degree, exp by base) and they're typically quick to be able to give the correct answer in a variety of situations. It's not until I try to teach them the rigorous justification that they start to get confused and everything slows down. So maybe it's "bad math", but there isn't a big difference between the two in the problem you gave. – Aeryk Nov 15 '15 at 20:20
• I have never in my life seen this use of infinity in calculus work. What an abomination! – Daniel R. Collins Nov 16 '15 at 4:20
• On the other hand, if you take the limit laws as axioms (without any attempt to justify them), you can compute these examples from the single fact that $1/x \rightarrow 0$ as $x \rightarrow \infty$. I don't personally see this as a particularly compelling way to introduce students to limits, but you could then explain this solution in a single class meeting. It should be clear that the first solution does not fit the given axiomatic framework, but I don't see any reason why students would understand why the given axiomatic framework is preferable to one based on rules to manipulate $\infty$. – Michael Joyce Nov 17 '15 at 14:51
• As a real-world example of student mistakes when employing this type of reasoning, I've encountered the problem on $\lim_{x\rightarrow\infty}e^{x+1}/e^x$, with students simplifying to $e^\infty/e^\infty$. See matheducators.stackexchange.com/questions/10067/… . – Ben Crowell Nov 22 '15 at 21:59

## 5 Answers

I think it is too hasty to dismiss manipulation of "$\infty$" out-of-hand, although, yes, there is a widespread tendency among students to over-simplify, thus crossing various lines into trouble.

The first-presented version is not so bad as a heuristic, and might have been written by Euler or Lagrange. For that matter, it can be made more focused by $\lim{x-2\over x(x-1)}=\lim{x\over x^2}=\lim {1\over x}=0$, with or without the flourish of $\lim {1\over x}={1\over \infty}=0$ at the end. The latter is certainly the way to remember the conclusion, as opposed to "for all $\epsilon>0$ there exists $N$ such that for all $N'\ge N$..." which is too quantifier-rich.

I'd claim that there are two distinct issues here, apart from whether or not $\infty$ is a number, which should be separated. First, seeing that $x(x-1)\sim x^2$, or that $x-2\sim x$ (as $x$ becomes large), is an important skill, despite "asymptotic expansions" falling somewhat outside the contemporary version of "tradition". But this is very important in genuine mathematical, scientific, and engineering practice: seeing through the noise of details. Second, the comparison of a range of not-noisy functions as $x\to+\infty$: powers of $x$ versus exponentials versus logs, without the "$x-2$" noise to weary and confuse the novice.

If suitable use of "$\infty$" is short-hand for the latter, what's the problem? Ok, yes, students seem to be eager to corrupt things, but, ...

Edit: and the second choice is really not the most intuitive, and all the inverses suddenly create lots more symbols. Simplifying the asymptotics of both numerator and denominator first eliminates the need for this, and reduces the chances for student errors simply in the relevant elementary algebra, which I've found to be alarmingly high, perhaps the main bottleneck in doing the second sort of argument.

• Sorry, but -1 on this one. there is a widespread tendency among students to over-simplify The use of $\infty$ as a single symbol for all infinite quantities is already an oversimplification. Students have a hard enough time understanding calculus when we present it to them using a system that is logically self-consistent. We don't have any such system in which all infinities are the same size and can be notated the same way. We do have NSA, and there is a nice freshman calc book by Keisler that uses NSA: math.wisc.edu/~keisler/calc.html . Why do wrong what can be done right? – Ben Crowell Nov 16 '15 at 21:12
• @BenCrowell, not clear to me what "doing calculus right" would mean, nor that students can be induced to do the "more right" (more "logically correct") versions, for various reasons. Compelling heuristics seem to "move" non-specialist audiences, and many specialist/expert audiences, more than do "rigor", in many cases. For most audiences, I'd encourage active sanity-checking at all stages, rather than adherence to "rules/rigor". But I do recognize this is a stylistic preference, etc. – paul garrett Nov 16 '15 at 21:16
• As stated in my comment, doing it right means using a system that is not logically inconsistent. – Ben Crowell Nov 17 '15 at 16:53
• @BenCrowell, I certainly don't seek out logical inconsistency, but it seems to me that people in general have a better tolerance for ambiguity (say, in a not-formal system) than for careful attention to rules/formalism. In an informal "system" (or "non-system"), logical consistency or not are simply not well-defined... Since I myself do not think that "teaching logical precision" is the top priority in calculus, I do not see a mandate to make any formal system, but, instead, show how to do interesting things in an informal system. A choice. – paul garrett Nov 17 '15 at 17:16
• @BenCrowell: no one can pretend using a system which is not logically inconsistent. At least no one can prove such a claim. – Benoît Kloeckner Nov 24 '15 at 14:06

It might be enough to show an example or two of what can go wrong with the logic of the first solution:

$\lim_{x\to\infty} \frac{e^x}{x^2}=\frac{\infty}{\infty^2}=\frac{1}{\infty}=0$

But of course this is incorrect!

Here's one more, in case they don't know the exponential-polynomial comparison:

$\lim_{x\to\infty}\frac{x^2}{\sqrt{x+1}\sqrt{x+2}\sqrt{x+3}}=\frac{\infty^2}{\infty^3}=\frac{1}{\infty}=0$.

Wrong again!

• In the latter case, perhaps they would write the denominator as $\infty^{3/2}$, and the "logic" would carry through... – Benjamin Dickman Nov 16 '15 at 1:10
• Need to "disguise", e.g. with a logarithm, exponential, arctangent or so. Make clear it is not always easy to see what "power" of infinity is involved, so this won't work in general. – vonbrand Nov 16 '15 at 1:23
• $\lim_{x\to\infty} \frac{e^x}{x^2}=\frac{e^\infty}{\infty^2}=\infty$. In the worst case, one can simply always write $\infty$ instead of $x$ after $\lim_{x\to\infty}$. This method will work until there is another limit and disguising will not change it. One can even write things like $\frac{f(\infty)}{g(\infty)} \stackrel{H}{=} \frac{f'(\infty)}{g'(\infty)}$. I agree that it may result in misunderstandings in communication with mathematicians using standard notation. Probably it is getting more and more artificial and is less useful in most cases, but you cannot say that this method is wrong. – BartekChom Nov 24 '15 at 14:56
• Do see my answer for more examples of the type you've provided. =) – user21820 Nov 24 '15 at 16:04

In the spirit of paw88789's answer, first show:

$\color{red}{ 1 = \lim_{x\to\infty} 1 = \lim_{x\to\infty} \dfrac{x}{\sqrt{x}\sqrt{x}} \overset{???}{=} \dfrac{\infty}{\infty\infty} \overset{???}{=} \dfrac{1}{\infty} = 0 }$.

And ask the students what is wrong with it, and attempt to argue with them that this reasoning is correct. By forcing them to pinpoint the problem on their own, you automatically make them much better at avoiding the same erroneous thought process themselves.

Almost surely students will be able to 'point out' that the "$\infty$" on top is bigger than each of the individual "$\infty$"s below. Ignoring that $\infty$ isn't well-defined, pursue the matter and ask them, why so, and why should it make any difference? They should say that it was "$x$" on top but "$\sqrt{x}$" below, and that if you don't distinguish the 'infinities' then you get the incorrect conclusion that the top is 'smaller' than the bottom.

Then it is easy to now tell them that using a single symbol "$\infty$" has resulted in loss of information of how big $x$ is getting, so we simply cannot use "$\infty$". This should be enough to convince students of the futility in using the first 'method' in your example.

As for the correct solution, I do not recommend the one you propose but rather:

$\dfrac{x+2}{x(x-1)} = \dfrac{x(1+\frac{2}{x})}{x^2(1-\frac{1}{x})} = \dfrac{1}{x} \dfrac{1+\frac{2}{x}}{1-\frac{1}{x}} \to 0$ as $x \to \infty$.

With the following thought process:

Identify the significant terms in each subexpression. On the top, for large $x$ the $2$ is comparatively insignificant to the $x$, and at the bottom likewise. We want to see how fast it grows, so we isolate that main growth rate. On top it is $x$, and at the bottom it is $x^2$. As we can see, at the bottom the leftover is $1$ plus/minus something small, so indeed $x^2$ has the same significant part (digits) as the original denominator. After simplification, we are left with $\frac{1}{x}$ multiplied by something close to $1$ then divided by something close to $1$. Intuitively it is clear that the overall growth is significantly the same as $\frac{1}{x}$.

Why is this better? Because students get a real feel for the actual growth rate rather than attempting to blindly churn out the answer. Not only that, it generalizes readily without difficulty to asymptotic expansions, which are widely used in engineering, physics and chemistry but that students are hardly taught to be able to manipulate!

After practicing with many examples, it would be a good idea to show students the following error:

$\displaystyle \color{red}{ 1 = \lim_{n\to\infty} 1 = \lim_{n\to\infty} \sum_{k=1}^n \frac{1}{n} \overset{???}{=} \lim_{n\to\infty} \sum_{k=1}^n \frac{1}{\infty} = \lim_{n\to\infty} \sum_{k=1}^n 0 = \lim_{n\to\infty} 0 = 0 }$.

Again as before, it is best to draw the students into a discussion and try to 'convince' them that it is correct. It should become clear that not only is $\infty$ dangerous to try using, it is still very dangerous even if only the limiting variable is substituted by $\infty$! This idea is a very common but fatal one, as the following examples will show any students that remain stubbornly unconvinced that $\infty$ is not to be trifled with:

$\displaystyle \color{red}{ \lim_{n\to\infty} \sum_{k=1}^n \frac{k}{n^2} \overset{???}{=} \sum_{k=1}^\infty \frac{k}{\infty^2} \overset{???}{=} {???} }$.

And if they argue that $k$ is on average $\frac{1}{2} \infty$ in the above summation then show them:

$\displaystyle \color{red}{ \lim_{n\to\infty} \sum_{k=1}^n \frac{k^2}{n^3} \overset{???}{=} \sum_{k=1}^\infty \frac{(\frac{1}{2}\infty)^2}{\infty^2} \overset{???}{=} \infty \times \frac{\infty^2}{4\infty^3} \overset{???}{=} \frac{1}{4} }$. [FALSE!!!]

• Not related the the other comments above: My counter-intuition is that I wouldn't even want to be seen writing improper infinity symbols (your red error text above) because that will visually get stuck in students' heads. I'd rather directly say "infinity is not a real number, that is false, you will be graded on proper writing", and supplement by asking them to do some problems where their method will generate the wrong answer to spotlight the cognitive dissonance. Show only correct writing on the board. – Daniel R. Collins Nov 25 '15 at 18:38
• @DanielR.Collins: Yours certainly a valid approach, and I didn't say that teachers should show what I just did. But I'm answering the actual question, which is about how to show students why the first way in the question is wrong in contrast with the second way. This already means that students have some misconception with limits and infinity. Don't forget that this is almost certainly true, what with the utter nonsense on the internet concerning infinity like Numberphile. And I think it is good if the red stuff get stuck in students' heads so that they can avoid doing the same. – user21820 Nov 26 '15 at 2:53
• Comments are not for extended discussion; this conversation has been moved to chat. // I moved a lengthy chain of comments in its entirety to chat as there was no clear cut-off; please feel free to continue it there. I only preserved a pair of comments that are explicitly unrelated to this other exchange. – quid Nov 26 '15 at 9:31
• I would like to draw attention of the future readers to the fact that such arguments did not convince at least me. This notation can be used to get correct results in this cases as I wrote in the comments moved to chat and in another chat room. – BartekChom Dec 3 '15 at 13:11
• Please do not continue the line of discussion that was moved to chat here in comments. I left one comment by each of the main contributors. Everybody interested in debating this, please visit the chat-rooms. (cc @BartekChom ) – quid Dec 4 '15 at 9:50

Elaborating on some of my earlier comments:

To me, the difficulty is not so much with exposing what is wrong with method (1), but the challenge of having students genuinely understand method (2) beyond merely manipulating symbols.

To genuinely understand method (2), one should have (a) have a precise definition of limit, (b) understand how the limits laws follow from this definition, and (c) understand how to apply the limit laws in this problem along the lines of method (2).

Typically, when I teach calculus, I only attempt part (c) of this process, and students either don't appreciate why method (2) is reliable in a way that method (1) is not or (worse, in my opinion) accept the limit laws as some magic and completely unmotivated dogma that is introduced completely ad hoc. I do try to motivate the limit laws and explain that they can be proven using a precise definition of limit, even though we do not cover the details of this in the course. Still, I worry that this falls on deaf ears far too often.

In practice, the students who best understand the material typically get that method (2) works in all the types of problems that they see, whereas method (1) can lead to wrong answers and so is not reliable. Thus, students have an empirical basis for trust in method (2), but almost never a truly mathematical basis for belief in the reliability of method (2). Arguably, this is all that can be accomplished in a first introduction to calculus (at least in American universities similar to mine), given the practical needs to cover other material in the course. At the end of the day, calculus is fundamentally about rates of changes, accumulation, and the deep connection between these two ideas. Limits are necessary to fully understand these concepts and so need to be substantially developed, but not at the expense of preventing the students from spending an adequate amount of time studying the primary themes of the course.

In my opinion, the moment you write $\frac{1}{\infty}$ on the blackboard and treat it as a first-class mathematical object, you have lost. Students are going to misuse it.

Teach them that "$\lim_{x\to a} f(x) = \infty$" is a notation for an epsilon-delta statement meaning that $f(x)$ is unbounded, and that a theorem says that if $a(x)\to A \in\mathbb{R}$ and $b(x)\to\infty$, then $\frac{a(x)}{b(x)}\to 0$. You may then point out that $\frac{1}{\infty}$ is a shorthand to recall the theorem, or a slogan, but using it in actual calculations is just humbug.

• Who said anything about writing $\frac{1}{\infty}$ on the blackboard? You don't generally teach the students to do this stuff, they do it by themselves. – Jessica B Nov 25 '15 at 7:57
• @JessicaB Even better -- then it is easier to argue against it: "No one taught you this method, and we didn't because it doesn't work (show counterexample). Don't treat infinities as numbers." – Federico Poloni Nov 25 '15 at 9:13
• Have you seen students change their work in response to saying something like that? – Jessica B Nov 25 '15 at 9:16
• @JessicaB: I don't teach calculus, but all my tests (statistics, college algebra, etc.) require that one "justify with properly-written math". This is actually specified as the top goal in every course. If students write falsehoods and nonsense statements then they lose points, fail tests, and either fix the problem or drop out of the course. – Daniel R. Collins Nov 25 '15 at 18:33
• @JessicaB: I think we've found the problem. Maybe that would be an interesting top-level question to ask. – Daniel R. Collins Nov 26 '15 at 2:30