What is a better way to explain these claims about limit are not true in general?

Question

As a TA who led calculus* 1 and 2 discussion section and holds office hour** in the previous year, I heard the following (wrong) arguments several times.

1. $\displaystyle \lim_{x\to \infty} \sqrt{x+1}-\sqrt{x}=0$ because $\infty-\infty=0$.

2. $\displaystyle \lim_{x\to \infty} x^{1/x}=1$ because $\infty^0=1$.

3. $\int_1^{\infty}f(x)dx$ and $\int_1^{\infty}g(x)dx$ both diverge so $\int_1^{\infty}f(x)+g(x)dx$ diverge.

I usually explain the arguments are not true in general by providing a (very trivial) counter example, for example,

1. $\displaystyle \lim_{x\to \infty} f(x)=\infty$ and $\displaystyle \lim_{x\to \infty} g(x)=\infty$ does not guarantee $\displaystyle \lim_{x\to \infty} f(x)-g(x)=0$, for example, $f(x)=x+1$ and $g(x)=x$.

2. $\displaystyle \lim_{x\to \infty} f(x)=\infty$ and $\displaystyle \lim_{x\to \infty} g(x)=0$ does not guarantee $\displaystyle \lim_{x\to \infty} f(x)^{g(x)}=1$, for example, $f(x)=2^x$ and $g(x)=1/x$.

3. False in general, for example $f(x)=-g(x)=1$

After giving explanations like that I sometime heard "But in your examples you can cancel the expression/formula..." and I was not sure how to continue. I tried the following methods, non of them seem to work very well.

a. Provide a much more complicated counter example which requires a few minutes of calculation to get the answer. This often leads to further confusion.

b. Just say that is the wrong way to do it. It sounds like "I'm the teacher so believe me." and doesn't do too much.

c. Show them the correct way to do their problems. This is almost like b (Why is your way the right way and mine is the wrong way?).

I'm looking for a better way to deal with questions like these.

*$\epsilon-\delta$ definition is not introduced. ** Office hour is in tutoring center where I'm also responsible for students take the class from the professors I'm not TA'ing for.

Answer 1

This is a difficult problem, because the students you're dealing with aren't really prepared to think about logical reasoning and the role of counterexamples. (For example, I find students at that level often don't distinguish between examples and arguments.)

I'm of the view that you can't actually expect students to understand material at a level much deeper than it's being taught. If an algorithm works on all problems they'll see then, in a real sense, the algorithm is correct for purposes of the class.

As it happens, the examples you're describing are probably not correct, even "locally", because they're going to be expected to deal (if not immediately then later in the course) with problems for which that algorithm doesn't work, so you do need to correct these misconceptions.

My suggestion would be:

1. Focus on giving counterexamples. It's important to avoid trivial problems. Students treat algorithms as heuristics, and have a hierarchy from easier to harder heuristics; you have to choose problems which trigger the faulty heuristic, and on which it fails. There's a natural supply of these: the problems which students will later have to solve using more sophisticated methods (L'Hospital's rule problems for a and b, other integrals for c). You don't need to work out the solution; you can say "Okay then what's the solution to this problem?" and then when they get the wrong answer, say "Actually, we're going to learn later that it's really ...". Instead of doing the calculation, you can draw (or even better, use a computer to generate) a graph of the function which supports your answer (don't present it as a proof, of course, just emphasize that it's suggestive, and the rigorous method will come later).

2. Another meaningful kind of counterexample, if you can use it, is problems where the shortcut doesn't apply. In other words, don't give a counterexample to its accuracy, give a counterexample to its usefulness. You don't want to say "this is right because I'm the teacher", but it's reasonable to say "we need this method to solve hard problems, so we're going to practice it on easy problems, even if there's a shortcut".

Answer 2

Often in courses like this, the problem is that students are almost exclusively asked to do problems that involve rote computation using procedures they've been taught, not problems that require reasoning about the concepts involved or even constructing their own examples. So, how about assigning problems that require students to come up with examples refuting various false notions like the ones mentioned? (Or at least discuss problems like these in class.)

I'm thinking of problems like:

• "Give an example of functions $f$ and $g$ such that $\lim_{x \to \infty} f(x) = \infty$, $\lim_{x \to \infty} g(x) = 0$, and $\lim_{x \to \infty} f(x)^{g(x)} = 5$."
• "Give an example of functions $f$ and $g$ such that $\lim_{x \to \infty} f(x) = \infty$, $\lim_{x \to \infty} g(x) = \infty$, and $\lim_{x \to \infty} (f(x) - g(x))$ does not exist."

Answer 3

Wanted to write this as a comment, but I'm not strong enough [reputation]. This may be a bit heavy handed or you're already doing it, but what about challenging the very notion of arithmetic when dealing with infinities before going towards counterexamples? That's usually what the fuss tends to be about.

As an aside, when teaching probability theory, the Cauchy distribution tends to flummox students because we run into problems when trying to compute the mean even though the distribution is symmetric. Why does there have to be a central tendency? It challenges a notion that they've held to be implicitly true. When they can't answer this general question of "why", it often helps to break the cognitive dissonance and they are more receptive to the idea that, yes, perhaps the mean doesn't have to exist.

Similarly, with limits involving infinities, I would ask students what rule says that $\infty - \infty = 0$ every time? Then, trivial examples as you give can help to seal the deal and disavow them of such a notion. Perhaps? Basically and I'm not sure if you already do this, but I would first challenge, philosophically, what students have considered to be axiomatically / dogmatically true before showing counterexamples.

Answer 4

I think your first strategy, giving counter-example, is a good start; you may have more success by giving counter-examples that carry meaning and relate to things the students already know. In the case at hand, you can also try to make the point that limits are not about arithmetic, but about asymptotic behavior.

I will consider the first fallacy you mention for illustration.

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To give counter-examples that carry meaning, you can draw two graphs of functions, and have the student guess what their limits are (which you have taken to be $+\infty$). Then point (ask him or her to show you) on the graph the difference between the two functions: it is represented by the vertical space between the two graphs. Now you can draw examples where both functions go to $+\infty$, but the gap between them goes to zero, or $+\infty$, or any given number, or does not have a limit. Then you can ask (or explain how) to turn these drawings into precise counter examples. Hopefully, symbols will start to carry a meaning in the students mind rather than being purely abstract. One of the thing we teachers too often forget to tell to our students is the process that made us propose a counter example. We are like magicians pulling a rabbit out of our hat, while our true goal is to attract their attention to the trick. By the way, I prefer by far to use hand drawn graphs, because a computer generated graph needs you to think of the formula first, feed it to the computer, and then observe. What we really do is to think of the general form of the graph, and then come up with a formula. Also, students are far more likely to feel they can use this methods of reasoning themselves with hand drawn graphs than with a computer, hand you want them to be able to debunk their next fallacies themselves.

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To relate the fallacies with things they already know, you can ask them if it is true that when two (positive) functions go to $+\infty$, their quotient goes to $1$. They may know that the answer is negative, and you can start get them to explain it to you (to themselves, really). Then the refreshment can be used in the case at hand, which is in fact awfully close.

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To make the point that limits are not about arithmetic, but about asymptotic behavior, you can try to separate two things: the meaning of the limit (what is the behavior of the function in a certain point or direction?) and the tools to determine limits. I guess that student often confuse the two (in a wide array of topics, not only limits; as an example, many student end up considering eigenvalues as being defined as the roots of the characteristic polynomial, leaving them unable to prove that the vector with all entries $1$ is an eigenvector of the matrix with all entries $1$: I have had almost all of my students failing this question because they wanted to compute the characteristic polynomial first).

You can thus try to rephrase the first fallacy: "if two functions go to $+\infty$ as the argument goes to $+\infty$, then the gap between the two must go to zero". They may realize they have no reason to believe that (but the drawings above may be necessary to do that). Relate to the true fact that when the common limit is not $+\infty$ but a real number, then the difference must go to zero. Have them understand that the use of arithmetic is made possible then because in this situation, the conclusion is inevitable. Even if they don't have the tools to prove it, they can see it. There you are at the root of the problem (they applied a rule they learned in a situation it does not apply to, without seeing that there is a missing hypothesis; such reasoning by similarity is very common). Once they realize the difference between the two situations (finite and infinite common limit), they may accept to at least doubt what they first said. The confusion might have been entrenched in their mind because they have seen the cases when one of the two functions goes to $\infty$ and the other has a finite limit, so make them look at this case two (without formulas at first, just graphs).

Answer 5

The first thing I would say is that principally you want to stick with strategy (a), and not veer into (b) or (c). You definitely want to dig into, and spend time on, these trouble spots with proper mathematical reasoning; certainly not "Because I said to do it this way", as that doesn't actually count as math at all. Counterexamples are logically the fundamental way one knocks down conjectures like those, so students should exercise and be trained in that method of dialogue.

What I do in a situation like this is try to jot down those trouble spots and turn them into quizzes for the whole class. Usually if I can turn it into (a) a binary "true or false" question, and (b) get the whole class in on the discussion, then we very quickly come to a consensus on the correct answer.

If you're solely working in a tutorial/one-on-one situation, then this becomes trickier to implement. Perhaps write down the statement as a "true or false" question and try to pin the student down to the one correct response (no "but this case was different" allowed). Possibly leverage one or two other nearby students to get their input on the discussion.

Answer 6

I find it in general a bit backwards if a teacher has to explain why such and such claim isn't true. For any given statement that that can't be proven, the default assumption should be that it's not true^†.

The entire confusion could be avoided by never talking about “infinite results” in the first place. If you replace $\lim f(x) = +\infty$ with simply the limit doesn't exist, then it's not possible to forge that into an unsound expression like $\infty^0$.

Now, of course what we don't want to do is make it more difficult to arrive at correct results, such as l'Hospital. But IMO, it's actually helpful if one doesn't start with infinite limits of some $f$ and $g$ for such theorems: the theorem is all the more impressive if it doesn't just give you a way to calculate some particular result, but actually “generates” that result, which you previously had no reason to assume existed!

This kind of proof-oriented thinking was never really explained to me at school, and I only started really enjoying maths when I understood it at university.

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^†_{Of course, one should not assume that it's wrong either. It's often good not to think in a true/false dichotomy...}