What's wrong with l'Hopital's rule?

Question

Upon looking at yet another question on this site on evaluating a limit explicitly without l'Hopital's rule, I remembered that one of my professors once said something to the effect that in Europe (where he is from) l'Hopital's rule isn't "overused" like it is here in the USA.

My question is, is there some reason not to use l'Hopital's rule when you have an indeterminate form? I know other techniques but l'Hopital is certainly my go-to. Is there some reason for hostility toward l'Hopital's rule?

Answer 1

L'Hopital's is often described as "overused" because, students especially, often forget to check the conditions for L'Hopital's Rule. For example, if after using L'Hopital's the limit does not exist, then that says nothing about the original limit. L'Hopital's is often used before checking for other forms. E.g. if we try to evaluate a variation of the limit $$\lim_{x\to 0} \frac{\sin x}{x}$$ most students will use L'Hopital's, will make the process harder and longer. That's why many professors, especially oldschool ones, do not approve of using L'Hopital's rule too often and will often take off points for using it when it is uneccessary.

Answer 2

L'hopital's rule fails sometimes in the case of this example taken from [1.]; it cannot be used to evaluate the limit of \begin{align} \lim \limits_{x\rightarrow\infty} \frac{x}{\sqrt{x^2+1}}. \end{align}After applying L'hopital's once we get $$ \lim \limits_{x\rightarrow\infty} \frac{\sqrt{x^2+1}}{x}, $$ and then applying it again yields$$ \lim \limits_{x\rightarrow\infty} \frac{x}{\sqrt{x^2+1}}, $$ which we see will just indefinitely loop. I also agree with SalmonKiller that L'hopital's rule seems to just be an automated response by many; this is both a testament to its usefulness and a major pitfall, as there are times for which L'hopital's rule is not efficient.

[1.] Weisstein, Eric W. "L'Hospital's Rule." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/LHospitalsRule.html

Answer 3

I can only guess that your professor feels it is an overpowered tool. Consider the limit $$\lim_{x\to 0} \frac{\sin(x)}{\sin(x)+\tan(x)}$$ There is a lot to be gained for a math student by spending more time on this limit. After all, the numerator is approaching zero, so it seems that the limit could be zero. The denominator is also approaching $0$ so maybe the limit doesn't exist. Clarity comes when you multiply through by $\frac{\csc(x)}{\csc(x)}$ The limit is $1/2$, which is actually sort of curious! (to me at least when I first learned it) It demonstrates a peculiarity of limits in that they can defy intuition. Mathematicians should experience this regularly. You might lose this sort of thinking experience if you just apply L'Hospital's rule off the bat. There are also limits like $$\lim_{n \to \infty}\left(1+\frac{x}{n}\right)^n$$ that as a first year calculus student I would not be able to do without L'Hospital's rule. Even using L'Hospital's rule I would have found that to be a tricky limit. I'm guessing your professor ultimately feels that L'Hospital's rule should be used as a last resort, when calculating the limit through other means is just not realistic (at that students current level of ability). Otherwise, the student should really spend time contemplating the limit and try to coax out the answer with some critical thinking. I know a number of my college professors felt this way about students using Tabular integration when doing integration by parts. I was a huge fan of Tabular integration because it made integrals like $$\int x^4\sin(x)\text{d}x$$ a breeze. Eventually I had a professor intentionally write all tests and quizzes so that Tabular integration wouldn't apply. This forced me to get a lot better at integration by parts, and that skill was truly a blessing when I later started learning about Fourier Series. If you continue on in mathematics you will get to real analysis. You will do limits much more rigorously, and L'Hospital's rule will probably not be available for use. The intuition you gain now from tangling with limits will benefit you down the road.

Answer 4

Students love L'Hopital because it's a magic wand they can wave over a problem, leaving intuition and actual thought undisturbed. A lot of students, at least in the U.S., don't understand the proof of the rule (it's not a trivial proof), and indeed don't really understand the definition of the derivative. Many problems that can be zapped by the magic wand are actually derivatives in disguise, and it's a shame if L'Hopital is invoked for these, since the student loses the opportunity to reconnect with one of the most important definitions in mathematics. My opinion: It's even overused on MSE (although I'm a fairly new member, I've seen a lot of it here). Example: We see things like $\lim_{n\to \infty} (\ln n)/n^2 = 0$ proved by L'Hopital on these boards. There's something wrong with that, again in my opinion, and I do thank you for reading my rant.

Answer 5

(Edit: I focus on the case where $\underset{a}\lim f=\underset{a}\lim g=0$ (ie $f(a)=f(g)=0$ by extending $f$ and $g$ by continuity). But the other cases can be reduced to this case by considering $1/f$ and $1/g$)

It is worthwhile to remark that one of the only cases where the subtlety of L'Hopital's rule is actually needed is when $f$ or $g$ has a vertical tangent at $a$ (ie $f'$ or $g'\underset{a}{\rightarrow}±\infty$), which is rare. For instance : $\lim_{x\rightarrow1^-}\frac{\arccos(x)}{\sqrt{1-x}}$

In (almost) all other cases, when $f'(a)$ and $g'(a)$ exist and $g'(a)≠0$, the result is a mere consequence of the very definition of differentiability (namely, $f$ is differentiable at $a$ ⟺ $f(x)-f(a)\underset{x\rightarrow a}\sim (x-a)f'(a)$ ), since

$\frac{f(x)}{g(x)}=\frac{f(x)-\overbrace{f(a)}^{=0}}{g(x)-\underbrace{g(a)}_{=0}}\underset{x\rightarrow a}\sim\frac{(x-a)f'(a)}{(x-a)g'(a)}\underset{x\rightarrow a}\sim\frac{f'(a)}{g'(a)}$

(Edit: Even when $g'(a)=0$, it seems to me that L'Hopital's rule is often superfluous, because when $f$ and $g$ are differentiable enough times at $a$, one can apply Taylor's theorem to them, which is often faster than L'Hopital's rule which may require repeated derivatives.)

The beauty and subtlety of L'Hopital's rule stem from the fact that it doesn't require $f$ nor $g$ to be differentiable at $a$. Using this subtle and complex theorem in cases that are easily solvable with stronger assumptions and simpler properties is pedagogically absurd.

Edit: So, from my point of view, this is what explains the bad image of L'Hopital's rule: magical and often unproven so of little pedagogical value, and moreover quite useless compared to more efficient and easier to prove methods, except in some rare cases.

(It seems to me indeed that L'Hopital's theorem is especially popular in the U.S. curriculum compared to the rest of the world. For example, in France, it is totally absent from the curriculum, and is only marginally treated as an exercise. I find this much more consistent.)

Answer 6

I will go against the grain and say that no, there is no reason not to use l'Hopital's rule, in cases where the conditions for its use hold.

Claims of its "overuse," and frequency with which students are instructed "not to use l'Hopital's rule," puzzle me. (Especially when the intended solution is to Taylor expand the numerator and denominator---which is basically what l'Hopital amounts to!). One of the great beauties of mathematics is that a theorem, once proven, is ours to keep forever. If l'Hopital's rule is an extremely flexible mathematical tool that can solve a wide variety of problems, all the more reason to teach it and encourage students to become familiar with it.

Pedagogically, I understand the need for students to practice using other tools for evaluating limits, like the squeeze theorem etc. So... include problems in the homework that specifically require those tools (and where l'Hopital's fails)?