# The case for L'Hôpital's rule?

While this may seem very subjective — and, admittedly, my own dislike of L'Hôpital's rule is not entirely devoid of subjectivity — I am looking here for argumented, factual answers.

From what I understand, students in the United States, when learning calculus and limits, are provided with and encouraged to use L'Hôpital's rule very early on. As a result, based on e.g. the activity on Math.SE, the use of L'Hôpital's rule ends up being pervasive and somewhat of a reflex for many students.

Assuming for once that the application of L'Hôpital is well-justified, and that the assumptions are checked, etc., this is a valid technique. However, based on my limited experience in research, this is a technique I have never seen used in "real" life." I frequently see people rely on asymptotic equivalents, Taylor approximations, integral comparisons, etc; all of them tools that, AFAIK, are barely taught to high-school or undergrad students.

My question, thus, is: why? What is the rationale in promoting the (almost exclusive) usage of L'Hôpital's rule in secondary education (again, in the USA)?

• Are my observation wrong, and are other techniques actually emphasized as well? (based on the sample data I have, I sort of doubt it)

• Is there a clear advantage to teaching L'Hôpital's rule and training students to use it by default, advantage that I am missing? (e.g., either educational, or in terms of further studies/applications)

• – Clement C. Jun 5 '16 at 16:32
• This is a very good point. If you have time to waste, have a look at matheducators.stackexchange.com/questions/8339/…. I think that you could better ask the question at matheducators. In any way, I join the club. Cheers. – Claude Leibovici Jun 5 '16 at 16:37
• I fully support your point of view. This rule can be dangerous, because students forget there are hypotheses to be checked (they retain the music, not the lyrics…). As a student, I was told that when it works, Taylor at order $1$ works as well. – Bernard Jun 5 '16 at 16:57
• I tell my students that l'Hôpital can be compared to the Paris gun; it can do the work it's meant to, but often times it's difficult to get through the wreckage. – egreg Jun 5 '16 at 21:04
• I fully agree with you. L'Hospital has always been my last resort. (I also get quite a shock that Taylor isn't even taught to undergrads in U.S.. Is it really the case? That would be too bad.) – Vim Jun 6 '16 at 7:17

## 2 Answers

Its great that you raised this point as a question (+1)! The issue has regularly been discussed in comments to many questions/answers tagged limits but now there is hope that all the arguments of both types (for and against) can be found in one place.

There is a very common misconception that limits can be evaluated via plugging and use of L'Hospital's Rule even furthers this line of reasoning that limits can be evaluated by "differentiating and plugging" instead of just "plugging". This does make evaluation of some of the complicated limits very easy and is perhaps the main reason for promoting its usage in USA. I think educators in USA want to teach Calculus I (or AP Calculus) in a manner which is just an extension of algebra. Thus the focus in evaluating limits is on usual algebraic operations of $+,-,\times,/$ and $=$ instead of order relations. Students just need to know that if plugging does not work then differentiating and plugging will work. Also such an education emphasizes on "formal differentiation" where students are presented with the rules of differentiation (sum, product, quotient, chain rule etc) together with the differentiation formulas for elementary functions. Such an approach is especially suitable for those students who are taking calculus course from applied perspective ("engineering" students) who are not going to deal with the "rigor" part of calculus anyway.

The above approach is so much contrary to the very spirit of calculus which is essentially based on non-algebraic notion of order relations $<, >$ and completeness. It is essential to grasp the idea that limits are essentially different from value of a function and that they are used to study behavior of values of a function rather than dealing with one value of a function.

Students trained on this approach normally get totally stuck when they encounter $\lim_{x \to 0}x\sin(1/x)$ and that shows the extent of such an approach. Another common problem is that students equate L'Hospital's Rule to "differentiate and plug" and rarely focus on verifying the hypotheses under which it works. Some students don't even bother to check if the expression is an indeterminate form $0/0$ or not.

However there is one plus side of L'Hospital's Rule which I want to highlight. It is easier to teach L'Hospital's Rule as a technique of evaluating limits than teaching the more powerful technique of Taylor's series. The proof of L'Hospital's rule is easier compared to the proof of Taylor's Theorem. Plus the students need to be aware of manipulation of infinite series (multiplying, dividing and composing two infinite series easily at least for few terms) in order to effectively use the technique of Taylor's series.

In my opinion students should be taught all the techniques starting with rules of limits, Squeeze Theorem, standard limit formulas (like $(\sin x)/x \to 1$ as $x \to 0$), L'Hospital's Rule and Taylor's series and preferably in that order.

I would disagree with your premise that l'Hopital's rule is almost never used. Actually it is important in building up the basic framework of the calculus. For example, if you wish to establish typical limits for transcendental functions, the rule is useful. I would agree with you that more advanced applications require more advanced estimates.

• I would say LHR is somewhat indispensable in evaluating some special limits like $\ln x/x$, but beside this, it has very limited scope in practical use. So I believe it's important much more in the theoretical sense (well, if at all) than in the real world. – Vim Jun 6 '16 at 7:26
• Not that I want to nitpick (your answer provides a good educational motivation for learning this rule in order to establish other theorems), but my statement was not that L'Hopital "is almost never used." It is that it is almost never used (as far as I could tell) in research, and outside secondary education math classes.: "However, based on my limited experience in research, this is a technique I have never seen used in "real" life."" – Clement C. Jun 6 '16 at 16:05
• My point was that it is the foundation for other results that are certainly used all the time. It is too elementary to be used in its own right. – Mikhail Katz Jun 6 '16 at 16:11