Near-universal student mistake on $\lim_{x\rightarrow\infty}e^{x+1}/e^x$

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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).