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On a recent first-semester calculus exam, I gave a bunch of limits. The student was supposed to use L'Hospital's rule if possible, or if not, explain why it didn't work and evaluate it by some other method. One of the problems was this:

$\lim_{x\rightarrow\infty}\frac{e^{x+1}}{e^x}$

Roughly 90% of them asserted that the limit was equal to 1, usually with some variation on the argument that the $+1$ would become negligible compared to $x$ as $x$ got big. Only one student correctly found the limit to be $e$.

I was surprised by the near-universal repetition of the same mistake. I would be interested for others' insights on why this would happen. I suppose a contributing factor might be lack of facility with manipulating exponents. It would have seemed obvious to me (once I'd realized that L'Hospital wouldn't help) to simplify the expression inside the limit, and it also seems obvious to me that exponentials turn addition into multiplication. Probably they are also forming some kind of false analogy with limits of expressions such as $(x+1)/x$. One or two seemed to be using bogus symbolic reasoning with the $\infty$ symbol, simplifying the expression to $e^\infty/e^\infty$.

It also surprised me that so many students made mistakes on these that they could have easily caught simply by plugging in some sample numbers on a calculator, such as $e^9/e^8$ in this example. Maybe it would be helpful to assign homework in the future on which they were required to do this kind of check.

Related: How long would it take to teach proper limit calculations?

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    $\begingroup$ I would love to have a huge list of questions like this which, for whatever reason, show some common misunderstanding. $\endgroup$ – Steven Gubkin Nov 22 '15 at 22:34
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    $\begingroup$ I like your idea of a required numerical check. $\endgroup$ – Joseph O'Rourke Nov 22 '15 at 22:46
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    $\begingroup$ If anyone has an exam coming up that covers this topic, it might be interesting to embed this problem on it. We could see whether this is something specifically wrong with the instruction I'm giving. $\endgroup$ – Ben Crowell Nov 22 '15 at 23:20
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    $\begingroup$ Perhaps the most useful technique would be to teach students to graph the function. If they had done that, they would have seen that this function is constant. I myself always understood limits best, when looking at graphs. $\endgroup$ – Amy B Nov 23 '15 at 0:44
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    $\begingroup$ These kind of errors will always be commonplace whenever limits are taught without any usable definition in order to compute them. I think 95% of limit problems (some very high percentage certainly not all) could be solved trivially by students by plugging in some extremely large number (or some number extremely close to the value where the limit is being taken to) and seeing if there is some obvious number very close to the answer. This is not generally what you are trying to teach, so allowing this numerical computation would make this method perhaps too viable. $\endgroup$ – PVAL Nov 23 '15 at 19:05
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While all of your students at this point will have done (extensive) units on manipulating and simplifying expressions with exponents, this is the limits unit. When doing limits questions, most students are only searching their brains for limits techniques - they're extremely unlikely to come up with the $\frac{x^a}{x^b} = x^{a-b}$ rule that had been driven into them by rote some 3 years ago.

(see my question: Compartmentalization in K-12 curriculum)

Since the correct bit of reasoning isn't readily available, students are likely to substitute a question that they can answer for this one that they aren't sure about. The (rather beautiful) fact that $\frac{x+1}{x}$ goes to $1$ as $x$ grows is very fresh in their minds.

I'd say that many of your students had some idea that this fact didn't jive exactly with this question, but in an exam situation you've sometimes got to just go with what's immediately available to you.

Edit: see this paper by Doug Rohrer for some extended commentary on the 'induced blindness' that your students seem to have been suffering from on this question.

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    $\begingroup$ I would add that if simplifications like $\lim_{x\to\infty} \frac{(x+1)(x+2)}{x+3}=\lim_{x\to\infty} \frac{x^2}{x}$ are done in one step, the students may think that such simplifications are always allowed. $\endgroup$ – BartekChom Nov 23 '15 at 15:29
  • $\begingroup$ @BartekChom: Your example simplification is wrong, and that is why students make mistakes, because they aren't taught the proper way to manipulate limits. One should never throw away 'apparently negligible' terms. $\endgroup$ – user21820 Nov 24 '15 at 11:45
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First of all, I think that the problem statement can be confusing.

use L'Hospital's rule if possible, or if not, explain why it didn't work and evaluate it by some other method

It can be misleading for students because it is easy to assume that if a fairly complicated method does not work (L'Hospital's rule), what you should do is use an even more complicated method (otherwise, why should you use the complicated method in the first place ?). So students tend to fall into the following reasoning :

  • Try L'Hospital's rule
  • L'Hospital fails
  • Search for more complicated method
  • Do not find it
  • Panic
  • Get stuck into the washing machine
  • Answer by analogy
  • Answer 1

Anyway, it can not explain everything, that is why I introduced the concept of "answering by analogy". Some (most ?) students tend to survive in maths class without understanding the basic rules of mathematics or why it is important to follow them. How is it possible that a student, with false reasoning, gives a correct answer ?

Just use follow the following reasoning

  • Look at the problem
  • Search a formula, which looks like the problem, but for which you have a solution
  • Adapt the formula to make it fit your problem
  • Use the adapted solution as a solution to your problem

For many problems it will work just fine. It works very well for factorization for example. But sometimes it fails.

In your example, when they are searching for something looking like $\frac{e^{x+1}}{e^x}$, they naturally find $\frac{x+1}{x}$ in their mind (as mentioned by NiloCK). And if it looks like the same, it must behave the same, that is how mathematics work, isn't it ?.

In my experience it is a real issue, since this kind of practice is hard to spot, and generally when you spot it, it is hard to explain why a method that just worked fine for years suddenly break down.

Edited : conclusion rephrased and extended

To conclude on a more general remark, it feels very natural for me to teach as an absolute rule to always simplify everything, and only when it is done to start using more advanced methods. With this in mind, the statement of the problem seems really strange, since it asks to violated what I consider as an absolute rule.

I do understand the relevance of such a problem, for example to show that l'Hospital's rule does not always help, or to verify explicitly if students have understood a given method, but it can help answer why so many students fails at that particularly point.

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    $\begingroup$ Apparently this was in the section in which l'Hopitals rule is taught. It makes sense to see cases where it breaks down, and this is a brilliant example. Requiring them to see the breakdown, and then find another way forward, is great. Probably better for a homework than for an exam, but still a really great question. $\endgroup$ – Steven Gubkin Nov 22 '15 at 23:46
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    $\begingroup$ I was always teach to first simplify everything and only when it is done to start using more advanced methods. I am quite puzzled by the fact that it does not seem to be what you advise to your students. You seem to have leaped to this conclusion without any evidence. $\endgroup$ – Ben Crowell Nov 22 '15 at 23:57
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    $\begingroup$ @BenCrowell (Not that I necessarily agree with the conclusion, but) he quoted evidence for that position in his post. $\endgroup$ – NiloCK Nov 23 '15 at 0:42
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    $\begingroup$ @DanielR.Collins: I've done that. Kolaru seems to have assumed without evidence that I haven't. $\endgroup$ – Ben Crowell Nov 23 '15 at 20:52
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    $\begingroup$ @BenCrowell I mus admit that my last paragraph did sound like some sort of accusation, which was not my intention (and would indeed lack evidence to be relevant). So I rephrased my conclusion. $\endgroup$ – Kolaru Nov 24 '15 at 18:18
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I believe the problem is as usual, namely that your students have no proper logic foundation, so they just followed what they believe are rules for symbolic manipulation but without any real understanding of what those manipulations may mean, not to say when they are valid or invalid. In finding limits, there are two commonly taught but erroneous ways.

Error 1

One error is to use the "$\infty$" symbol in the midst of a computation. There is never any reason to use it. For example, always use:

$\dfrac{x^2+3}{4x^2-5} = \dfrac{1+\frac{3}{x^2}}{4-\frac{5}{x^2}} \to \dfrac{1}{4}$ as $x \to \infty$.

And never:

$\color{red}{\dfrac{x^2+3}{4x^2-5} \overset{???}{\to} \dfrac{\infty^2+3}{4\infty^2-5} \overset{???}{=} \dfrac{\infty^2}{4\infty^2} \overset{???}{=} \dfrac{1}{4}}$ as $x \to \infty$.

Because the latter is so very wrong in so many ways.

The best way is always to teach students to identify the significant terms and keep them in front in each subexpression, and manipulate the expression without changing the value in order to align the significant terms or reduce them. All this should be done without attempting to take limits, until the very end.

So for your example students should have done:

$\dfrac{e^{x+1}}{e^x} = \dfrac{e^x e}{e^x}$.

Simply because $e^x$ is the significant part of $e^{x+1}$ and it also aligns with the $e^x$ in the denominator.

Error 2

The other 'error' is to teach the limit laws as usually presented to students who are unable to understand the proofs completely. Instead of manipulating limits by reducing subexpressions to their limit values, students should be made to manipulate the expressions themselves without reducing anything but keeping enough information about everything.

Instead of:

$\color{red}{\lim_{x\to 0} (1+x^5)(2-3x^4) = \lim_{x\to 0} (1+x^5) \times \lim_{x\to 0} (2-3x^4) = 1 \times 2}$.

Which conveys almost zero meaning whatsoever, students should be taught to think:

$(1+x^5)(2-3x^4) \in (1+o(1))(2-o(1)) = 1 \times 2 + o(1)$.

Where you read each "$o(1)$" as "some little bit" and emphasize that the little bits may be different and may not be positive. And one should clearly explain why it is true:

$(1+o(1))(2-o(1)) = 1 \times 2 - 1 \times o(1) + o(1) \times 2 - o(1) \times o(1) = 2 + 4 o(1)$.

Because you have literally $4$ little bits there, whose sign you don't know, so they don't cancel one another.

At the end, when the limit is wanted, only then do we look at what is the significant term and ignore the little bits. The reason for doing this is so that the concepts and techniques extend naturally and easily to asymptotic expansions, where we want finer distinction between the little bits. (Some little bits are littler than others.)

Conclusion

With both of these in place, it is simply impossible for students to have any incorrect conceptual notions about limits, and the kind of error that you observed will never occur (unless they're not interested in learning).

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  • $\begingroup$ "Because the latter is so very wrong in so many ways." I am not fond of the first way, however better it is to a wrong way. One can use your later suggestion and teach asymptotic equivalence, so that one can correctly write $\frac{x^2+3}{4x^2-5}\sim_{x\to \infty} \frac{x^2}{4x^2} = 1/4$. $\endgroup$ – Benoît Kloeckner Nov 24 '15 at 14:16
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    $\begingroup$ As another comment, I am quite fan of Landau's notation $o(\cdot)$, but I find it very difficult to use with non-advanced students. Indeed, it looks like a function, but we write the same expression for different functions! In the same formula no less! Hugely difficult to grasp and manipulate correctly, until one is really comfortable with functions and a bit of analysis. $\endgroup$ – Benoît Kloeckner Nov 24 '15 at 14:18
  • $\begingroup$ @BenoîtKloeckner: Well concerning your first comment, we don't have experiments comparing my approach with your approach, but I do believe that students will grasp limits fully (100% understanding) in a shorter time using my approach. About your second comment, I don't care whether Landau's notation is used, but the same idea should be conveyed. If one wishes, one could use $\boxed{?}$ or something instead of $o(1)$. $\endgroup$ – user21820 Nov 24 '15 at 15:00
  • $\begingroup$ @BenoîtKloeckner: And if you prefer, we can include an intermediate step of $\dfrac{x^2+3}{4x^2-5} = \dfrac{x^2(1+\frac{3}{x^2})}{4x^2(1-\frac{5}{4x^2})}$ that is motivated by pulling out the significant part of each subexpression just like we think of significant digits. This intermediate step has exactly the same import as asymptotic equivalence but on cancelling the significant terms immediately gives the same expression as I suggested, which is the discrepancy factor. I've given a more detailed explanation of this thought process at matheducators.stackexchange.com/a/10077/1550. $\endgroup$ – user21820 Nov 24 '15 at 16:13

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