How long would it take to teach proper limit calculations?

Question

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.

Answer 1

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.

Answer 2

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!

Answer 3

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!!!]

Answer 4

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.

Answer 5

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.