# Why are people so interested in finding limits without l'Hôpital's rule?

It is not difficult to find questions about evaluating limits without the use of l'Hôpital's rule. As long as the function is differentiable, not directly reducible and tends towards an intermediate form, why would anyone want to avoid such a useful tool?

• Why are people so interested in finding limits using L'hospital's rule? After all, this technique is rather anecdotal and often complicates things. – Did Mar 10 '14 at 0:34
• @Did: In my experience, most of the questions where the asker explicitly excludes L'Hospital are ones where it would have greatly simplified the evaluation of the limit. Just because it's not the right tool for everything is no reason to refuse to use it for tasks that it is the right tool for. – hmakholm left over Monica Mar 10 '14 at 15:30
• Vibhav, if the expression does not take off to an intermediate form, there would be no motivation to use lhospital's rule in the first place! – Christian Chapman Mar 11 '14 at 14:47
• I completely agree with Did. I never used l'Hopital's rule, and in fact I don't even know what it says. The so-called "développements limités" (in French; that is, Taylor's formula with the "little o" remainder) are usually so efficient to compute limits that I don't see any reason for using something else. – Etienne Mar 12 '14 at 19:23
• So Gerry's answer would be OK for a question like: "why do people want to find limits without using Taylor's formula"... – Etienne Mar 12 '14 at 21:55
• One nice thing about the limit definition is that it always works, plus it applies in any dimension. L'Hopital's rule has (in my opinion) somewhat delicate conditions for it to hold, plus it fails almost completely to generalize to higher dimensions (since it relies on the Cauchy mean value theorem). Pedagogically speaking, I would hesitate in an analysis sequence to let students blindly use L'Hopital's rule without fully appreciating these two points. But I suppose there's not much of a choice in a calculus sequence. – Gyu Eun Lee Mar 14 '14 at 3:27
• @Did because if learned and applied correctly, it's a good tool. anecdotal and often are subjective terms. – TZakrevskiy Mar 16 '14 at 20:20
• @TZakrevskiy "Good tool" is a subjective term. "Often" could be made objective, using MSE questions as a database and counting how many times L'H was applied correctly, how many times it was necessary, and how many times it was the simplest approach. My feeling is that the figures are appalling (but this is a subjective appreciation since I did not do the stats). – Did Mar 16 '14 at 22:06
• @Etienne This is an artifact of your French background where the developpements limites are introduced before power series and instead of L'Hopital's rule. This implies that the problems chosen for the French problems set reflect this. In general, my assessment is that there is a large class of "easy" problems that are equally simple with both methods because they simply involve the derivative, and then there is a class of problems that is easier with the asymptotic approach because this permits to develop different parts of the expression to different precision, – Phira Mar 16 '14 at 22:55
• @Phira I again completely agree with Did. Still, I would be glad to know a non-artificial example where L'Hopital's rule can be applied very easily whereas asymptotic expansions cannot, or only at the cost of rather complicated manipulations. – Etienne Mar 17 '14 at 11:48
• @Did a non-crafted example where L'H works better than Taylor expansion is finding $\lim_{x\to\infty}\frac{\ln x}{x}$. – TZakrevskiy Mar 17 '14 at 20:56
• @TZakrevskiy This one is good! – Etienne Mar 18 '14 at 10:03
• @ParamanandSingh Instead of two trivial differentiations for L'H you write plenty of inequalities - and one needs to find the correct inequalities first! Hardly an efficient way. As I said, we can use swords to spread butter, but table knives do it better. – TZakrevskiy Dec 14 '14 at 11:17
• @TZakrevskiy: I agree with your last line. But in my view "LHR=swords" and "Squeeze theorem=table knives" and you can compare these techniques on the basis of complexity of the proof of each of these theorems. But I agree that whenever LHR works it is a very good/efficient/powerful tool. I prefer however to use powerful tools when such power is needed (like for $\lim\limits_{x \to 0}\dfrac{x - \sin x}{x^{3}}$) – Paramanand Singh Dec 15 '14 at 4:16
• @TZakrevskiy: I know this is an ancient post, but I want to point out that I have no problems with students using L'Hopital's rule iff they can on their own prove it rigorously. If they cannot even state it correctly (with all the precise conditions), I have a serious problem with the teacher. – user21820 Sep 22 '17 at 5:28

There are occasions when the use of l'Hopital is circular. E.g., when using l'H on $$\lim_{x\to0}{\sin x\over x}$$ you have to differentiate $\sin x$, but to differentiate $\sin x$, you have to evaluate $\lim_{x\to0}{\sin x\over x}$.

• I wish someone told me this in Calculus :C – Prism Mar 10 '14 at 6:41
• @Ephraim How do you prove that $\frac{d}{dx}\sin x=\cos x$? By applying the definition of derivative? What do you end up with? – Daniel R Mar 10 '14 at 20:56
• @Ephraim, I didn't say it didn't work; I said its use was circular, as using it depends on knowing the very limit you are using it on. – Gerry Myerson Mar 11 '14 at 5:10
• Indeed, any use of L'Hopital's rule on a limit as $x\to0$ of $f(x)/x$ (in that form), where $f(0)=0$, is by definition circular. – Glen O Mar 11 '14 at 9:02
• @Glen It may be redundant, but not necessarily circular. If the derivative of $f$ is already known, there is no circular reasoning here. It's true that the answer could be also obtained simply by recalling the definition of derivative, but this does not invalidate the L'H approach. – user127096 Mar 14 '14 at 19:18
• @127.0.9.6 - the problem is that the derivative of $f$ is defined by the limit, and therefore the fact that it is "already known" is really just another way of saying "someone has previously already worked out the limit". – Glen O Mar 15 '14 at 15:43
• @Glen The definition as a limit is not the only way to find derivatives: there are product, chain rules, etc. – user127096 Mar 15 '14 at 16:03
• @127.0.9.6 - yes, but all of those methods of finding derivatives are derived, ultimately, from the definition of the derivative itself - that is, $\lim_{h\to0} \frac{f(x+h)-f(x)}h$. – Glen O Mar 15 '14 at 16:23
• I think the circular proof of that limit depends on the definition of $\sin$, if $\sin$ is defined as the ratio of the opposite side to the adjacent side then the proof becomes circular. But $\sin$ can be defined in many other way for example define $\arcsin x$ as an integral and $\sin$ as it's inverse although such definitions are not popular, in that I think case l'hospital can be used to prove the limit. A common example would be the limit $\lim_{x\rightarrow 0}{\frac{\ln{1+x}}{x}}=1$. Many books starts with defining $\ln {x}$ as an integral and using l'hospital to prove the limit. – hrkrshnn Mar 16 '14 at 11:28
• @Hurkyl - the fact that you can recover the differentiation operator by starting with the assumption of things like the chain and product rules really is irrelevant. The concept of the derivative comes from the physical idea of determining the instantaneous gradient of a curve. Furthermore, the proof of L'Hopital's rule depends on the limit definition of derivative, and therefore existence of other definitions is rather irrelevant. – Glen O Mar 17 '14 at 2:41
• @Glen: That may have been the original and motivating application, but the whole reason derivatives are useful is because of their simple algebraic manipulations. Limits of difference quotients are something you might tack on at the beginning or the end if relevant to the problem: it's not something you imagine you're using throughout. I don't believe that limits were even part of the original calculation method: I recall Newton did something more analogous to algebraic treatments of formal derivatives, such as the use of dual numbers. – user14972 Mar 17 '14 at 8:43
• @GerryMyerson Not necessarily circular, even with the right triangle definition. It's not really true that you need first principles to differentiate $\sin$. For example, you can use vector calculus to prove that $\left (\sin(x)\right )' = \cos(x)$ by only the symmetries of unit circle. – Balarka Sen Mar 23 '14 at 10:09
• @BalarkaSen: and in particular that vector calculus proof, not l'Hopital, tells you what $\lim_{x \to 0} \dfrac{\sin x}{x}$ is. – Robert Israel Mar 23 '14 at 23:49
• I don't see any circularity. It's like if I prove the quadratic formula in full generality, then use it to solve the roots of $x^2+4x+4$; yes, I already have to know how to find the roots of squared monomials to prove the quadratic formula, but that does not mean I can't do the above calculation! Using L'H on your limit is only circular if the student is attempting to use it as a step in a proof that $\sin'(x) = \cos x.$ – user7530 Dec 22 '14 at 18:43
• TFW a meta post turns into a math post – Christian Chapman Feb 21 '16 at 6:41

Personally, I'm against any calculus technique that can be applied without (much) thinking. These days calculus (in North America, at least) is taught in a way that people can get high grades without having the slightest idea of what a derivative or an integral is. In most classes I teach I ask what an integral is, and very rarely do I get satisfactory answers, even from good students. Part of the problem is the lack of basic skills: most students are hopeless when dealing with inequalities, which prevents you from both explaining the definition of limit and doing things like Taylor polynomials.

The way I was taught calculus a million years ago, was to use Taylor (as opposed to L'Hôpital) for limits. Using Taylor approximations to find the limit allows you to have some understanding of what is going on, in particular in the sense that you are not only finding the limit but also estimating the rate of convergence. This is essential if you are doing numerical analysis, and good knowledge in any case.

This conveys more information, makes you think instead of blindly applying a formula, and avoids mistakes like the frequent one of applying L'Hôpital when it is not applicable.

• Hmm.. the L'Hospital's rule I know boils down to neither more or less than simultaneously working out the Taylor expansions on both sides of the fraction bar one term at a time, until you hit a nonzero coefficient in one or both of them. (Well, modulo factors of $n!$ that cancel out anyway). It's not clear to me that there's any real dichotomy here. – hmakholm left over Monica Mar 12 '14 at 0:37
• I personally always hated the "what is an integral" type questions. To me, an integral is whatever it needs to be in order to solve a problem. Hell, there is still research being done on how differential calculus needs to be defined. I feel no obligation to conceptualize differentials the way some one other person does, or to be limited to their definitions. -- I much prefer questions that require an understanding of an integral to solve. "A bathtub is filling with water at a certain rate...." – DanielV Mar 13 '14 at 12:27
• Whether you like it or not, convention is an essential ingredient in mathematics. And integration is a notion that goes well beyond solving some calculus problems. The question I often ask is "what do the symbols $\int_a^bf(t)dt$ mean?" The answers I get are frequently very vague, and that's a problem if the class where that happens is an advanced analysis one. These days students can pass a calculus class with high grades and not be able to answer the question above, which I find ridiculous. – Martin Argerami Mar 13 '14 at 12:55
• @DanielV You sound like a physicist. Heresy I say! Thumbs up for having the same sentiment as Newton. – kleineg Mar 13 '14 at 17:54
• @MartinArgerami What is this way of teaching? I am interested. – Billy Rubina Mar 13 '14 at 23:18
• @Pristine: basically, what I advocate is that calculus (or any math, for that matter) should be taught with the goal of understanding the concepts as opposed to be doing mindless calculations (which can be easily done by Wolfram Alpha). – Martin Argerami Mar 14 '14 at 4:59
• @Martin Argerami: I'm curious: do you count the answer to that question toward the grade for your classes? In other words, if they can't answer it, they don't get to pass or pass with high grades. – The_Sympathizer Mar 15 '14 at 22:09
• Not usually. I asked the question in classes after calculus. My point is that I usually find students who got 95% in calculus but cannot answer that question. And I'm not saying it is their fault, it is ours for the pathetic way in which we teach calculus. – Martin Argerami Mar 15 '14 at 23:56
• It is the whole point of algorithms that they can be used without thinking... – Phira Mar 17 '14 at 10:06
• If you want to make sure that your students have understood things, then you should just also ask questions that check for understanding and that cannot be solved with common algorithms. – Phira Mar 17 '14 at 10:11
• @Phira: have you ever tried teaching a section of a multi-section first year class in your "own way"? – Martin Argerami Mar 17 '14 at 11:30
• Here: math.stackexchange.com/questions/716663/… is an example of the attitude I'm talking about. – Martin Argerami Mar 18 '14 at 11:06
• @HenningMakholm For limits as $x\to\infty$ it does not boil down to Taylor. – user127096 Mar 18 '14 at 17:34
• @DanielV: "there is still research being done on how differential calculus needs to be defined" Really? Can you cite an example? I honestly thought this was pretty much settled a century ago. Unless you're about talking mathematical logic and non-standard approaches involving infinitesimals... – Jesse Madnick Mar 24 '14 at 0:01
• So given that the calculus teaching now is no good, then where do you get that understanding? I take it that knowing the formal definitions is not enough to explain "what an integral really is" (even if you know how to apply them well), so where do you get that from? – The_Sympathizer May 15 '15 at 22:43

In some circumstances questions like "How do I do X without Y" are genuinely intellectual exercises in working without powertools, but in other circumstances they seem more like "I have an aversion to thinking about Y so let's just do it another way."

My impression is that the first group is by far the bigger group in general, but for l'Hopital's rule specifically, it might be a mix.

Anyhow, this idea of not relying on a single route to a solution can be viewed as a positive development in the student's development :) Many students, when finishing a problem through whatever means, would just conclude "Welp, good thing I never have to think about that ever again! No chance that any portion of that problem would ever help out in a future problem because all math problems are totally disconnected and don't relate to each other or reality. It's not as if there are similar problems where the same approach won't work, requiring me to find an alternate path."

Ok they wouldn't think all of this consciously, but really that's how it seems they think sometimes...

Anyhow, the positive upshot is that a student who is used to/recognizes the value of finding alternate solutions will be more flexible in the long run.

• +1 for "Welp, good thing I never have to think about that $\cdots$ all math problems are totally disconnected and don't relate to each other or reality. – Jay Mar 10 '14 at 22:37

This is a bit of a meta answer, but let me explain a bit of why a course might not cover l'Hôpital's rule. When I teach calculus I've skipped l'Hôpital for two reasons.

First, understanding when using it is and isn't circular is much more difficult than anything else covered in a Calc 1 class. Gerry gave a great example of a subtle circularity, but there are others. Since students won't be able to understand when they can and can't use it, they shouldn't use it at all.

Second, in my experience, learning l'Hôpital's rule causes students to forget everything else they ever learned about limits. In particular, many students will apply it to limits which are not indeterminate! Thus teaching l'Hôpital's rule causes more unlearning than learning, so I'd prefer to spend that time teaching another topic instead.

Now you might wonder why a class that never taught l'Hôpital's rule would have students who knew l'Hôpital's rule. When I've taught calculus usually a substantial portion of the class has taken a high school calculus class where they were taught l'Hôpital's rule but don't understand it. So I then have to give a brief explanation of why I'm not teaching it and why they shouldn't use it on the problems in the class.

• Which begs the question of when and how this technique was introduced and became so proeminent in the curricula of some countries (but not all) that the possibility of not covering it must be carefully justified. – Did Mar 10 '14 at 14:45
• @Did I think it's often taught because it's a way to formalize the intuitive idea that the value of indeterminate limits depends on how "fast" the nominator and denominator go to zero or infinity. Unfortunately however, it's not taught that way, but instead presented as a one size fits all solution, as Noah observed. – fgp Mar 11 '14 at 1:38
• Gerry's example isn't really a problem. You are given a limit, you see it's of the form f (x) / g (x), both have a well-known limit of zero and a well-known derivative... The fact that someone didn't realise that with sin (0) = 0 the limit sin (x) / x is by definition sin' (0) is not a problem. If I can solve 100 problems following a pattern (which obviously has to be applied with care), and don't notice that one of these had a more basic solution, where's the problem? – gnasher729 Mar 14 '14 at 13:53

Mainly because when you first study limits, you are not introduced to L'Hôpital's rule. Most limit questions come from beginners, who have not even studied derivatives. When the time comes to study L'Hôpital's, your interest in limits are generally boiled away. Hence, it is quite natural to see people asking limit questions without L'Hôpital's [perhaps because I have been through that stage].

And also, because many people see it as a challenge to find the limit without advanced techniques.

• So it's fundamentally a question of mathematical culture. In the culture in which I learnt mathematics, l'Hôpital's rule was taught before the formal epsilon-delta definition of a limit! – Peter Taylor Mar 10 '14 at 11:13
• The fact that L'H rule is explicitly mentioned in those questions suggests the users have heard of it. – user127096 Mar 10 '14 at 14:16
• @127.0.9.6: It was probably "covered" in their high school Calculus class. – Noah Snyder Mar 10 '14 at 14:53
• @127.0.9.6 Why not? You do have bit of knowledge what you may be studying ahead. And even if you see few good questions of limits here, you will know there is some stuff called L'Hôpital's, which allows people to evaluate limits easily. – Sawarnik Mar 10 '14 at 20:22

I'm pretty sure I've asked such a question, actually. The simple reason is that I was trying to solve an exercise, which was recommended to do in connection to a class which preceded the one where L'Hopital's rule was introduced. So I concluded that while the exercise was probably solvable using L'Hopital's rule (which I wasn't familiar with at that point), it was most likely intended to be solved by other means. And those 'other means' were what I was interested in, and not the limit itself.