Utilizing it bypasses some of the skills that are taught early on in calculus 1; by disallowing it, professors can get a better idea of whether students are truly learning and understanding the fundamentals.
It is not necessarily circular reasoning. It depends on your definitions and how you proved sin'(x)=cos(x). Circular reasoning is only bad in the first derivation. Circular reasoning is acceptable and useful in consistency checks, applications, and recall of known facts.
Isn't the example at the top of [the Wikipedia page for l'hopital's rule](https://en.wikipedia.org/wiki/L'H%C3%B4pital's_rule) incorrect? Neither f(x) nor g(x) have derivatives equal to zero at x=0.
The problem is that you need to know the limit of sin(x)/x as x->0 in order to get to the derivative of sin(x) in the first place. It comes up when using the limit definition of the derivative. So using the derivative of sin(x) to help evaluate that particular limit is bad logic. Of course it gives the correct answer *because it has to*, but it's sort of like saying 1+1=2 because 2-1=1
Of course, this is all assuming you're using the usual development of sine and its derivative. Some pedant always comes in to point out that this isn't a valid argument if you define sin(x) in terms of its taylor series.
I always considered it a cheat code in high school exams.. :)
Wish I would have learned the fundamentals better though. The uni hit a bit harder when I actually had to build on the fundamentals that weren’t as solid as I had thought. (STEM)
also better understanding wtaf you're actually doing = you're more likely to use that rule in more accurate places
if you think of it as just "ooh super easy way to get a fast answer" = you're almost definitely going to use it in the wrong place
I believe it is mainly to do with the fact it simplifies too many things bit like a cheat code if you would, doesn’t necessarily help you improve the foundations and understanding more
Students often don't even bother explaining why the theorem holds for the particular limit (both numerator and denominator are differentiable on an open interval containing the limit point C, except possibly at C itself, and that the denominator is not 0 except possibly at C).
So acknowledge it?
How am I, a person grading these, to be sure whether you understand the theorem or just learned a simple trick to differentiate the numerator and denominator?
This isn't a complicated theorem, even "the assumptions for l'Hopital's hold" is enough.
It's a similar case with double integrals where students won't bother to even write "this integral is equal to an iterated integral because of Fubini's theorem"
The big issue with l'hopitals rule is that it relies on derivatives, which are defined in terms of limits, and some of the things that students want to use it on are the limits that are used to define those derivatives (looking at sin x / x). This causes a loop in the logic that breaks the validity of the answer.
In order to be consistent, you need to find other ways to prove those limits.
not really a Cauchy-Weirstrassian or Newtonian derivative yes. A Hudde-Descartes derivative(adequality and power rule on a series definition of a function)or a Lagrange-Taylor(n! times the coefficient of x\^b an independently derived power series) However as Suzuki and Kaufman note youre limited to analytic elementary functions with those methods(in fact Suzuki points out the Hudde Descartes doesnt apply to transcendental functions and Im still struggling to justify chain rule in Hudde) and proving that these derivatives and Cauchy Weirstrass or Caratheodory deriviatives are the same object is a pain.
I agree on your observation, I'm currently studying here in the Philippines and our curriculum in Calc 1 doesn't allow us to use L'Hopital's Rule strictly prohibited. We were allowed to use it until it is taught conceptually.
Former high school math teacher here. From personal experience, students tend to apply it without checking for applicability and it sometimes confuses them with the similarity in application to the quotient rule. The other part is that as someone else said, l'Hopital's rule can lead to circularity without doing some algebraic work. Probably the most important skill that you should be boosting in your HS calc class is your algebra. Students would rather differentiate a radical than rationalize, but they are more likely to make an error trying to do a chain rule in their head than rationalizing. Then there is the simplifying that they would need to follow up with, often l'Hopital's rule improperly applied can give you some nasty algebraic expressions to simplify.
TLDR Version;
1. Confusion with the quotient rule
2. Algebraic manipulation would make the limit easy to evaluate already
3. Students don't check for applicability of l'Hopital's rule, they just "use" it
4. Algebraic manipulation is needed to make the l'Hopital's rule work
It's a bit of a sledgehammer for all your limits questions. Sometimes, you want to learn something by picking away elegantly at a problem rather than just smashing it to pieces.
However, in an exam situation, I would start with it to get the answer and then work out how to get the answer using just algebra.
> It's a bit of a sledgehammer for all your limits questions.
True, but students still need to learn how to use it 'safely'. It requires care, but it's too powerful and useful for applications (and theory too) to ignore.
On some exams, they are trying to see that you can use certain rules to do certain things. It's not about establishing a result, it's about applying rules.
It's sort of like someone solving trig identities by converting all the expressions to complex exponentials. It's valid math and probably simpler, but it's not what they want you do, in terms of running this particular obstacle course.
A lot of times, L’h is circular reasoning. Though there are ways to define stuff good enough without the circular reasoning problem (like defining some functions with its power series), but still, in general, the problem exists because you cannot (easily) redefine (all) the functions already defined in an exam, or you have to prove the definitions are equivalent, which is not easy sometimes or is omitted most of the times.
There are scenarios when L’Hôpital’s isn’t logically sound, but get the right answer anyways. There are lots of examples of that in math. But basically, when calc students learn derivatives, they are defined in terms of limits. But many students will use this rule on a limit that is defining a derivative. This is called circular logic and is a mathematical fallacy.
I don't think teachers don't like it. It is useful and has an interesting history. It is annoying when students use it wrong, use it on very simple examples, or use it when asked not to. Deborah Hughes Hallett was annoyed enough to leave it out of her calculus book. She was rightly ridiculed and included it in later editions.
You should use L'hopital's rule as a last resort. For many functions, you can use inequalities, do manipulations, or leverage some other relationship to do the limit without L'hopital's. Doing it these other ways demonstrate more mathematical skill and set you up better going forward. Being able to get equations into beneficial forms or to use inequalities to simplify things without messing with results are WAY more complex and important skills than just doing a couple derivatives.
Imagine someone at the gym using a forklift to lift a weight.
This is how a teacher sees a student when using L'Hôpital's rule. You don't calculate the limits just to know the limit, you're doing it to train your brain muscles. If you use L'Hôpital's rule, you're getting the result without putting in the effort, and that’s not the purpose of the exercise.
Speaking just from an AP calc standpoint, there are questions that are referencing the definition of the derivative at a certain input value. Those same questions can be answered using L’Hopital’s rule which to me misses the point of the question. I wish the concept wasn’t in the AP calc curriculum to help students build understanding of derivatives.
Nope. You would still have to to prove every derivative formula you use L'Hopital's on, without L'Hopitals. So you still have to do the algebra to evaluate limits that you are trying to avoid.
I think that’s silly. However, it’s also my opinion that every step taken should have its own justification. It just helps track where students went wrong and builds a better foundation for more rigorous mathematics
It is a useful tool. However, in math class, students are taught a variety of tools, and they need to demonstrate understanding of those tools, too.
Part of the issue is that using L'Hospital's rule can "bypass" other tools for limits, which is undesirable if you are teaching people various techniques and want them to demonstrate them.
For a simple example:
lim(x->infinity) of ( 2x\^3 + 3x + 5 ) / ( x\^3 + 2x\^2 )
If I want students to show they can divide by x\^3 / x\^3 to get:
( 2 + 3/x\^2 + 5/x\^3 ) / ( 1 + 2/x )
and then show that the other terms go to zero, I'd like them to do that.
If someone says "I just took derivatives until I had 12/6 = 2" they might not ever understand the other technique.
Similarly, in calculus, learning the difference quotient is important for understanding of the derivative. Very few people will keep finding derivatives that way, but it's useful as a learning exercise. So if someone was told to use the difference quotient to find the derivative of x\^3, just using the power rule would give the "right" answer, but not accomplish the actual goal of the exercise.
I'm really confused reading all these responses. L'Hopital's Rule is an easy shortcut student use to skip having to use harder rules? Why didn't I ever figure that out? All I ever learned as is this tool for edge cases.
well.. that rule has quite a lot of restrains. in Russian math school there is a philosophy: you have to prove, that L’hospital rule can be in use. of course almost every time you can, but still.
After having done Calc 1 to 3, I might have used L’Hôpital like…thrice, and never in any exam. Only ever use it as a “last resort”. As mentioned by most, OP, you will benefit wayyy more from learning how to factor properly. Factorisation will come in handy, later on. It’s what will make you save precious time. Especially when you’ll reach integration. It can make the difference between a simpler, 5 minute, Simplified integral, or a longer process, e.g. applying integration by parts more times to solve one integral.
I'm pretty sure I could amputate a man's leg with a bazooka. I'm also pretty sure we shouldn't train surgeons that this would be among the best ways to do that task, at least in a whole lot of situations.
Can the students confidently prove the rule? E.g. by referring to the Rolle's theorem in the right way? If they can, they sure as hell can compute an exam limit using some other method and won't even be bothered. If they can't prove it, it's fine to ban it, because they don't deserve it with their low understanding.
So we proved by exhaustion it's okay to not allow it qed.
This thread is so full of shit. Engineers, computer scientists, and other professionals in STEM don't need to be able to prove L'hopital's rule, any more than Mathematicians need to be able to build high-quality, performant applications, or do empirical research that moves the needle in science. The status quo in American universities whereby other people in STEM do less proof-oriented Math courses is correct and we should hold to it.
Also, there are proofs in Linear Algebra that would be beyond many Math professionals without seriously pursuing further studies. Many Integration Bee or Olympiad-style challenges would be unsolvable for many Math professionals without seriously pursuing further studies. And guess what? They don't need to know those things. We live in a world of specialization, and we ought to. People are going to pursue their own narrow area and stay in their lane. That's how the 21st Century works, and the only way it really can work.
Your first argument doesn't hold to scrutiny. You say it's useless in real life scenario to be able to do proofs. But then again, so is finding the limit of a function in the first place, because you can just ask wolfram alpha or whatever to find it for you in a fraction of a second. If anything, the reasoning abilities you would develop by proving things might prove more useful than an essentially mechanical skill.
The fact that you say the status quo of American universities (and you are right to mention it, as other countries indeed have engineers do proof based math) is right doesn't make it so, and only proves you are unwilling to, or unable of, changing your mind.
As for your second argument, the fact that even mathematicians can't do all the proofs that exist, means that STEM people should do none of them how, exactly?
I never said, "it's useless in real-life scenarios to be able to do proofs." I said it's unnecessary for other professionals in STEM.
Your critique of my first argument has an assumption embedded in it that is incorrect. You said "If anything, the reasoning abilities you would develop by proving things might prove more useful than an essentially mechanical skill." This assumes that writing mathematical proofs is the only way to develop reasoning skills, and this is patently false. Problems in Comp-Sci are a completely different animal, and people who are the best at solving tough coding challenges, and writing and debugging code in production may not know much at all about mathematical proofs. Similarly, people from a Math background are often not the best at comp-sci related things.
There is something very important to note here. Excelling at one task that requires reasoning is decoupled from excelling at another task that requires reasoning. We have theories as to why this is, but not solid theories. And it's also very important to note that it would have been hard to predict that this is the case with humans from first principles. Empirical evidence shows it to be true though. The best chess players, the best computer programmers, and the best mathematicians are rarely the same people. They almost certainly share underlying skill sets, but they have to devote hours to studying very different things.
They have to get good at reasoning, and maybe even "insight" (although it's not at all clear that insight is quite as important to other fields as it is to mathematics, and it's not clear that insight is trained or manifests the same way in comp-sci as it would in mathematics).
Everything I've just said incidentally reveals another very, very important point; reasoning can only get you so far in predicting outcomes. It's too easy to make mistakes reasoning from first-principles. Which is why science is driven by empiricism and any notion that it should disregard evidence in favour of reasoning from first-principles is bunk.
From first principles, someone may have predicted that "proof-writing skills will translate to all other fields that require formal logic!" But they clearly don't.
For this reason careful, systematic observation is obviously an essential component of other STEM fields and such empiricism is very far removed from mathematics. It is an essential skill that needs to be trained separately.
I will also point out that a "mechanical skill" is very ill-defined. What is a mechanical skill? This likely means following some steps as opposed to having "insights". All of that is ill-defined. For one thing, a great deal of applied science simply hinges on knowing the right thing to do in the right situation. It's about staying up to date with the literature and utilizing best practices in any given situation. Much of training ML models, and being a data science practitioner is like this. "Insight" or lateral thinking seems to be very helpful in software development (with the caveat again, that it's poorly defined), but for the most part, comp-sci people develop the insight and lateral thinking they need through a much different curriculum. A much different route.
My second point doesn't prove any of what I'm saying, it's a very useful example that is consistent with what I'm saying. Empirical evidence proves what I'm saying. Reasoning, using formal logic, (and even poorly defined "insight") depend heavily on specific tasks, and modalities and specific training is required within those tasks and modalities.
>This assumes that writing mathematical proofs is the only way to develop reasoning skills, and this is patently false.
Not at all. What I did say is that it "might" prove more useful, so nothing as definitive as what you're saying, and certainly nothing I wrote suggested that it was the only way to develop such skills. I was merely pointing out that just knowing how to find the results is at least as useless when computers can do it in a fraction of the time.
As for the rest of what you're saying, I mostly agree. I will say 2 things though.
First, anecdotally, when i was studying physics I found the skills required in both disciplines to not be so distanced. You're overdoing the "math is pure logic and far removed from reality" thing. I think there's a lot of empirism in math. I can't prove much if I'm not able to draw, visualize and have some "empirical evidence" to find patterns. And yes, it's possible to do that in more ares of math than just number theory or euclidean geometry. I've never seen any evidence for non mesurable sets, though.
That's not even to mention that a lot of math, even very advanced, was developed with the aim to model reality or because of a physical problem in the first place.
Also, again you saying that the curriculum of a computer scientist or other scientist is much different and doesn't have proof based math because they don't need it is not the slam dunk you think it is. Instead, it is merely an American centric point of view. If it's a proof of anything, it's a proof that you are maybe a bit narrow minded.
As mentioned in the other answers, using L'Hopital's rule allows students to get to the answer without properly understanding what they are doing, and in some cases using it would amount to circular reasoning. And even if it is later allowed, it's i.m.o. still better to use Taylor expansions with the order term for the error. So, to compute the limit of sin(x)/x for x to zero, we could say that in a neighborhood of x = 0, we have:
sin(x) = x + O(x\^3)
Therefore:
sin(x)/x = 1 + O(x\^2)
And Taylor expansions a provide for a far more efficient way to go about computing limits than using L'Hopital anyway, because you can use the known series expansions of the standard functions. Particularly if you have to use L'Hopital repeatedly, the calculational efforts ends up reproducing series expansion terms that you already knew in advance or could have computed in a far more efficient way.
Finally someone said it. Honnestly I don't understand the obsession with L'Hospital online when Taylor expansions are far more efficient. You just need to know the first terms of the expansions of elementary functions then you can find virtually any limit involving them. I was genuinely surprised to find that it was the go to method for so many people. In my education L'hospital was a footnote because we learned about Taylor expansions and it was clear they were much more efficient.
Sometimes there are better ways to find a limit (for example using a Taylor polynomial, or trying to bound the limit by some quantity). So if you just use L’Hopital’s rule your are missing on all those tools that are also valuable .
No student in math should use techniques ifthey don’t understand how it works. If you cannot prove it, you cannot use it.
I insist on my students finding the derivative as a limit of the difference quotient. Many smart ones know the differentiation rules and always argue with me. I make them derive the rules using the binomial theorem before letting them use the rules.
Edit: Wo! lookie here, I am getting downvoted for recommending students learn and understand proof of advanced concepts before using it—as if it’s bad advice. Peace out, folks! I am not wasting my time as a veteran math professor in this community.
I'm really glad my calculus 1 teacher insisted (against the school's guidelines) on us learning the concept of limits, before going on to showing us the definition of a derivative, before showing us some rules for derivatives and why they work. But "no student in math should use techniques if they don’t understand how it works" is just ridiculous. We constantly ask people to learn rules they can't prove. I'm not going to insist a first grader read *Principia Mathematica*.
We were discussing the l’Hôpital rule and why some teachers don’t approve of its usage. And I am talking about learning math for high school students and about my philosophy for my college students.
I am not talking about teaching pedagogy for grade school.
You don’t need the implicit function theorem to prove l’Hôpital’s rule. They need to at least appreciate the graphical sketch using the mean value theorem.
Math is not learnt properly without proofs and not all proofs at every level need to be rigorous but a proof is required.
At the college level, yes, I prove that theorem for my class.
In any case, that theorem is not needed at lower levels and any derivatives (like ln x etc.) can be proved using implicit differentiation.
Utilizing it bypasses some of the skills that are taught early on in calculus 1; by disallowing it, professors can get a better idea of whether students are truly learning and understanding the fundamentals.
There’s also an issue with circular reasoning that can come up. A famous example of this is using the rule on sinx/x as x->0
It is not necessarily circular reasoning. It depends on your definitions and how you proved sin'(x)=cos(x). Circular reasoning is only bad in the first derivation. Circular reasoning is acceptable and useful in consistency checks, applications, and recall of known facts.
I’m obviously talking about the way sine and its derivative are discussed in the overwhelming majority of books and classrooms
majority = right and only
Isn't the example at the top of [the Wikipedia page for l'hopital's rule](https://en.wikipedia.org/wiki/L'H%C3%B4pital's_rule) incorrect? Neither f(x) nor g(x) have derivatives equal to zero at x=0.
>Neither f(x) nor g(x) have derivatives equal to zero at x=0. Which is exactly why the rule works after one application of it.
The first example I see is f=sin x and g =-0.5x, which is definitely indeterminate
Yes but neither have a vanishing derivative at 0 (where the rule is being applied).
Oh I misread your question- the derivatives don’t need to be zero. The original function values do.
Oh you're right, my mistake.
What's wrong with this? It gives the correct answer no? (cos(0)/1 =1)
The problem is that you need to know the limit of sin(x)/x as x->0 in order to get to the derivative of sin(x) in the first place. It comes up when using the limit definition of the derivative. So using the derivative of sin(x) to help evaluate that particular limit is bad logic. Of course it gives the correct answer *because it has to*, but it's sort of like saying 1+1=2 because 2-1=1 Of course, this is all assuming you're using the usual development of sine and its derivative. Some pedant always comes in to point out that this isn't a valid argument if you define sin(x) in terms of its taylor series.
Ah, this takes me back to the Calc 1 days of solving derivatives by definition. Very informative, thank you!
Thats also what my prof told me, thats why we learned a Lot before He showed us L'hospital
diiiiiiiiiiiiiitto - that was the LAST thing we learned when talking about differentiating
I always considered it a cheat code in high school exams.. :) Wish I would have learned the fundamentals better though. The uni hit a bit harder when I actually had to build on the fundamentals that weren’t as solid as I had thought. (STEM)
also better understanding wtaf you're actually doing = you're more likely to use that rule in more accurate places if you think of it as just "ooh super easy way to get a fast answer" = you're almost definitely going to use it in the wrong place
Can you give an example of a problem where such a student might use it in the wrong place?
honestly no but I meant rules like that in general across the board
I believe it is mainly to do with the fact it simplifies too many things bit like a cheat code if you would, doesn’t necessarily help you improve the foundations and understanding more
Students often don't even bother explaining why the theorem holds for the particular limit (both numerator and denominator are differentiable on an open interval containing the limit point C, except possibly at C itself, and that the denominator is not 0 except possibly at C).
thats obvious bro
So acknowledge it? How am I, a person grading these, to be sure whether you understand the theorem or just learned a simple trick to differentiate the numerator and denominator? This isn't a complicated theorem, even "the assumptions for l'Hopital's hold" is enough. It's a similar case with double integrals where students won't bother to even write "this integral is equal to an iterated integral because of Fubini's theorem"
The big issue with l'hopitals rule is that it relies on derivatives, which are defined in terms of limits, and some of the things that students want to use it on are the limits that are used to define those derivatives (looking at sin x / x). This causes a loop in the logic that breaks the validity of the answer. In order to be consistent, you need to find other ways to prove those limits.
You can derive that limit without derivatives so using L’H to determine sin x/ x is valid
Using squeeze theorem on this limit is so satisfying.
Me when f'(c) = lim (f(x) - f(c))/(x - c) as x→c is simply a consequence of l'Hôpital's rule
That limit is used to find the derivative in the first place
You can find the derivative without knowing that limit
The derivative of sinx at x=0 is, by definition, lim x to 0 sinx/x So if you know the derivative, you already know the limit
Can you do so with highschool level calculus
The limit of sin(x)/x as x goes to 0? Yes, in fact there's multiple videos on Khan Academy about it.
No the derivative of sinx without the limit you mentioned
https://youtu.be/HVvCbnrUxek?si=mwGDa4PW8CTORQzN this uses the limit, but the limit is also proved: https://youtu.be/5xitzTutKqM?si=TBsUG_Sws9XoCGCJ
I think you misunderstood what I was talking about.....
no you need the Lagrange and Madhava Bhaskara II methods which today arent taught until a history of Math course or Galois/field theory course
You can't, "derivative" is a meaningless term without a concept of the limit.
Mathematicians from 1736 to 1821 punching the air rn
not really a Cauchy-Weirstrassian or Newtonian derivative yes. A Hudde-Descartes derivative(adequality and power rule on a series definition of a function)or a Lagrange-Taylor(n! times the coefficient of x\^b an independently derived power series) However as Suzuki and Kaufman note youre limited to analytic elementary functions with those methods(in fact Suzuki points out the Hudde Descartes doesnt apply to transcendental functions and Im still struggling to justify chain rule in Hudde) and proving that these derivatives and Cauchy Weirstrass or Caratheodory deriviatives are the same object is a pain.
Kids these days have really forgotten the lost art of trolling.
Without using power series, which require the derivatives to verify?
Power series do not require derivatives to verify if one chooses to define a function by its power series
Sandwich Theorem for days (yummy)
I agree on your observation, I'm currently studying here in the Philippines and our curriculum in Calc 1 doesn't allow us to use L'Hopital's Rule strictly prohibited. We were allowed to use it until it is taught conceptually.
Former high school math teacher here. From personal experience, students tend to apply it without checking for applicability and it sometimes confuses them with the similarity in application to the quotient rule. The other part is that as someone else said, l'Hopital's rule can lead to circularity without doing some algebraic work. Probably the most important skill that you should be boosting in your HS calc class is your algebra. Students would rather differentiate a radical than rationalize, but they are more likely to make an error trying to do a chain rule in their head than rationalizing. Then there is the simplifying that they would need to follow up with, often l'Hopital's rule improperly applied can give you some nasty algebraic expressions to simplify. TLDR Version; 1. Confusion with the quotient rule 2. Algebraic manipulation would make the limit easy to evaluate already 3. Students don't check for applicability of l'Hopital's rule, they just "use" it 4. Algebraic manipulation is needed to make the l'Hopital's rule work
It's a bit of a sledgehammer for all your limits questions. Sometimes, you want to learn something by picking away elegantly at a problem rather than just smashing it to pieces. However, in an exam situation, I would start with it to get the answer and then work out how to get the answer using just algebra.
> It's a bit of a sledgehammer for all your limits questions. True, but students still need to learn how to use it 'safely'. It requires care, but it's too powerful and useful for applications (and theory too) to ignore.
On some exams, they are trying to see that you can use certain rules to do certain things. It's not about establishing a result, it's about applying rules. It's sort of like someone solving trig identities by converting all the expressions to complex exponentials. It's valid math and probably simpler, but it's not what they want you do, in terms of running this particular obstacle course.
General principle: if its easier to run around the obstacle instead of going over it, you designed the course poorly.
There are always other ways to get to the finish line of an obstacle course, but avoiding the obstacles will not be considered passing.
That is lazy design.
I'm sure they care what you think. Especially since this seems like a pointless quibble.
Kinda like using the quadratic formula and skipping learning how to factor
A lot of times, L’h is circular reasoning. Though there are ways to define stuff good enough without the circular reasoning problem (like defining some functions with its power series), but still, in general, the problem exists because you cannot (easily) redefine (all) the functions already defined in an exam, or you have to prove the definitions are equivalent, which is not easy sometimes or is omitted most of the times.
There are scenarios when L’Hôpital’s isn’t logically sound, but get the right answer anyways. There are lots of examples of that in math. But basically, when calc students learn derivatives, they are defined in terms of limits. But many students will use this rule on a limit that is defining a derivative. This is called circular logic and is a mathematical fallacy.
Because L'Hôpital bought his theorems from Bernoulli. Makes them mad.
I don't think teachers don't like it. It is useful and has an interesting history. It is annoying when students use it wrong, use it on very simple examples, or use it when asked not to. Deborah Hughes Hallett was annoyed enough to leave it out of her calculus book. She was rightly ridiculed and included it in later editions.
Some insurance doesn’t cover trips to l’hopital. ;-)
I wish I had heard this back when I was teaching cal 1 lol xD
The same reason calculators aren’t liked in elementary school. It leads students to take shortcuts instead of learning the important concepts
You should use L'hopital's rule as a last resort. For many functions, you can use inequalities, do manipulations, or leverage some other relationship to do the limit without L'hopital's. Doing it these other ways demonstrate more mathematical skill and set you up better going forward. Being able to get equations into beneficial forms or to use inequalities to simplify things without messing with results are WAY more complex and important skills than just doing a couple derivatives.
Imagine someone at the gym using a forklift to lift a weight. This is how a teacher sees a student when using L'Hôpital's rule. You don't calculate the limits just to know the limit, you're doing it to train your brain muscles. If you use L'Hôpital's rule, you're getting the result without putting in the effort, and that’s not the purpose of the exercise.
Speaking just from an AP calc standpoint, there are questions that are referencing the definition of the derivative at a certain input value. Those same questions can be answered using L’Hopital’s rule which to me misses the point of the question. I wish the concept wasn’t in the AP calc curriculum to help students build understanding of derivatives.
this is true, I am still serving 4 more years of a 12 year sentence for using it in my homework :/
I wonder if proffesors would give you full marks if you proved it at the start of your paper then used it
Nope. You would still have to to prove every derivative formula you use L'Hopital's on, without L'Hopitals. So you still have to do the algebra to evaluate limits that you are trying to avoid.
Prove why it works first, then you can use it
I think that’s silly. However, it’s also my opinion that every step taken should have its own justification. It just helps track where students went wrong and builds a better foundation for more rigorous mathematics
It is a useful tool. However, in math class, students are taught a variety of tools, and they need to demonstrate understanding of those tools, too. Part of the issue is that using L'Hospital's rule can "bypass" other tools for limits, which is undesirable if you are teaching people various techniques and want them to demonstrate them. For a simple example: lim(x->infinity) of ( 2x\^3 + 3x + 5 ) / ( x\^3 + 2x\^2 ) If I want students to show they can divide by x\^3 / x\^3 to get: ( 2 + 3/x\^2 + 5/x\^3 ) / ( 1 + 2/x ) and then show that the other terms go to zero, I'd like them to do that. If someone says "I just took derivatives until I had 12/6 = 2" they might not ever understand the other technique. Similarly, in calculus, learning the difference quotient is important for understanding of the derivative. Very few people will keep finding derivatives that way, but it's useful as a learning exercise. So if someone was told to use the difference quotient to find the derivative of x\^3, just using the power rule would give the "right" answer, but not accomplish the actual goal of the exercise.
No l’hopital? That’s so sad :(( I’ve always used it since the beginning of calc 1 other than the one or two times we had to do it the long way
I'm really confused reading all these responses. L'Hopital's Rule is an easy shortcut student use to skip having to use harder rules? Why didn't I ever figure that out? All I ever learned as is this tool for edge cases.
well.. that rule has quite a lot of restrains. in Russian math school there is a philosophy: you have to prove, that L’hospital rule can be in use. of course almost every time you can, but still.
After having done Calc 1 to 3, I might have used L’Hôpital like…thrice, and never in any exam. Only ever use it as a “last resort”. As mentioned by most, OP, you will benefit wayyy more from learning how to factor properly. Factorisation will come in handy, later on. It’s what will make you save precious time. Especially when you’ll reach integration. It can make the difference between a simpler, 5 minute, Simplified integral, or a longer process, e.g. applying integration by parts more times to solve one integral.
I'm pretty sure I could amputate a man's leg with a bazooka. I'm also pretty sure we shouldn't train surgeons that this would be among the best ways to do that task, at least in a whole lot of situations.
Because l'hopital didnt discover it, he just put it in a bundle with other formulas
Can the students confidently prove the rule? E.g. by referring to the Rolle's theorem in the right way? If they can, they sure as hell can compute an exam limit using some other method and won't even be bothered. If they can't prove it, it's fine to ban it, because they don't deserve it with their low understanding. So we proved by exhaustion it's okay to not allow it qed.
This thread is so full of shit. Engineers, computer scientists, and other professionals in STEM don't need to be able to prove L'hopital's rule, any more than Mathematicians need to be able to build high-quality, performant applications, or do empirical research that moves the needle in science. The status quo in American universities whereby other people in STEM do less proof-oriented Math courses is correct and we should hold to it. Also, there are proofs in Linear Algebra that would be beyond many Math professionals without seriously pursuing further studies. Many Integration Bee or Olympiad-style challenges would be unsolvable for many Math professionals without seriously pursuing further studies. And guess what? They don't need to know those things. We live in a world of specialization, and we ought to. People are going to pursue their own narrow area and stay in their lane. That's how the 21st Century works, and the only way it really can work.
Your first argument doesn't hold to scrutiny. You say it's useless in real life scenario to be able to do proofs. But then again, so is finding the limit of a function in the first place, because you can just ask wolfram alpha or whatever to find it for you in a fraction of a second. If anything, the reasoning abilities you would develop by proving things might prove more useful than an essentially mechanical skill. The fact that you say the status quo of American universities (and you are right to mention it, as other countries indeed have engineers do proof based math) is right doesn't make it so, and only proves you are unwilling to, or unable of, changing your mind. As for your second argument, the fact that even mathematicians can't do all the proofs that exist, means that STEM people should do none of them how, exactly?
I never said, "it's useless in real-life scenarios to be able to do proofs." I said it's unnecessary for other professionals in STEM. Your critique of my first argument has an assumption embedded in it that is incorrect. You said "If anything, the reasoning abilities you would develop by proving things might prove more useful than an essentially mechanical skill." This assumes that writing mathematical proofs is the only way to develop reasoning skills, and this is patently false. Problems in Comp-Sci are a completely different animal, and people who are the best at solving tough coding challenges, and writing and debugging code in production may not know much at all about mathematical proofs. Similarly, people from a Math background are often not the best at comp-sci related things. There is something very important to note here. Excelling at one task that requires reasoning is decoupled from excelling at another task that requires reasoning. We have theories as to why this is, but not solid theories. And it's also very important to note that it would have been hard to predict that this is the case with humans from first principles. Empirical evidence shows it to be true though. The best chess players, the best computer programmers, and the best mathematicians are rarely the same people. They almost certainly share underlying skill sets, but they have to devote hours to studying very different things. They have to get good at reasoning, and maybe even "insight" (although it's not at all clear that insight is quite as important to other fields as it is to mathematics, and it's not clear that insight is trained or manifests the same way in comp-sci as it would in mathematics). Everything I've just said incidentally reveals another very, very important point; reasoning can only get you so far in predicting outcomes. It's too easy to make mistakes reasoning from first-principles. Which is why science is driven by empiricism and any notion that it should disregard evidence in favour of reasoning from first-principles is bunk. From first principles, someone may have predicted that "proof-writing skills will translate to all other fields that require formal logic!" But they clearly don't. For this reason careful, systematic observation is obviously an essential component of other STEM fields and such empiricism is very far removed from mathematics. It is an essential skill that needs to be trained separately. I will also point out that a "mechanical skill" is very ill-defined. What is a mechanical skill? This likely means following some steps as opposed to having "insights". All of that is ill-defined. For one thing, a great deal of applied science simply hinges on knowing the right thing to do in the right situation. It's about staying up to date with the literature and utilizing best practices in any given situation. Much of training ML models, and being a data science practitioner is like this. "Insight" or lateral thinking seems to be very helpful in software development (with the caveat again, that it's poorly defined), but for the most part, comp-sci people develop the insight and lateral thinking they need through a much different curriculum. A much different route. My second point doesn't prove any of what I'm saying, it's a very useful example that is consistent with what I'm saying. Empirical evidence proves what I'm saying. Reasoning, using formal logic, (and even poorly defined "insight") depend heavily on specific tasks, and modalities and specific training is required within those tasks and modalities.
>This assumes that writing mathematical proofs is the only way to develop reasoning skills, and this is patently false. Not at all. What I did say is that it "might" prove more useful, so nothing as definitive as what you're saying, and certainly nothing I wrote suggested that it was the only way to develop such skills. I was merely pointing out that just knowing how to find the results is at least as useless when computers can do it in a fraction of the time. As for the rest of what you're saying, I mostly agree. I will say 2 things though. First, anecdotally, when i was studying physics I found the skills required in both disciplines to not be so distanced. You're overdoing the "math is pure logic and far removed from reality" thing. I think there's a lot of empirism in math. I can't prove much if I'm not able to draw, visualize and have some "empirical evidence" to find patterns. And yes, it's possible to do that in more ares of math than just number theory or euclidean geometry. I've never seen any evidence for non mesurable sets, though. That's not even to mention that a lot of math, even very advanced, was developed with the aim to model reality or because of a physical problem in the first place. Also, again you saying that the curriculum of a computer scientist or other scientist is much different and doesn't have proof based math because they don't need it is not the slam dunk you think it is. Instead, it is merely an American centric point of view. If it's a proof of anything, it's a proof that you are maybe a bit narrow minded.
I genuinely read that as IHOP's rule. Context: IHOP is a pancake restaurant chain in the US.
As mentioned in the other answers, using L'Hopital's rule allows students to get to the answer without properly understanding what they are doing, and in some cases using it would amount to circular reasoning. And even if it is later allowed, it's i.m.o. still better to use Taylor expansions with the order term for the error. So, to compute the limit of sin(x)/x for x to zero, we could say that in a neighborhood of x = 0, we have: sin(x) = x + O(x\^3) Therefore: sin(x)/x = 1 + O(x\^2) And Taylor expansions a provide for a far more efficient way to go about computing limits than using L'Hopital anyway, because you can use the known series expansions of the standard functions. Particularly if you have to use L'Hopital repeatedly, the calculational efforts ends up reproducing series expansion terms that you already knew in advance or could have computed in a far more efficient way.
Finally someone said it. Honnestly I don't understand the obsession with L'Hospital online when Taylor expansions are far more efficient. You just need to know the first terms of the expansions of elementary functions then you can find virtually any limit involving them. I was genuinely surprised to find that it was the go to method for so many people. In my education L'hospital was a footnote because we learned about Taylor expansions and it was clear they were much more efficient.
I believe that this obsession mostly comes from the fact that Americans represent 50% of Reddit's userbase.
Yeah thanks Sherlock, but the question then becomes why Americans do it like that in the first place.
Sherlock would say that the answer lies in the question...
Sometimes there are better ways to find a limit (for example using a Taylor polynomial, or trying to bound the limit by some quantity). So if you just use L’Hopital’s rule your are missing on all those tools that are also valuable .
That’s generally how it goes because you have to learn the hard way before learning the easy way, it’s like that for derivatives and integrals too.
No student in math should use techniques ifthey don’t understand how it works. If you cannot prove it, you cannot use it. I insist on my students finding the derivative as a limit of the difference quotient. Many smart ones know the differentiation rules and always argue with me. I make them derive the rules using the binomial theorem before letting them use the rules. Edit: Wo! lookie here, I am getting downvoted for recommending students learn and understand proof of advanced concepts before using it—as if it’s bad advice. Peace out, folks! I am not wasting my time as a veteran math professor in this community.
I'm really glad my calculus 1 teacher insisted (against the school's guidelines) on us learning the concept of limits, before going on to showing us the definition of a derivative, before showing us some rules for derivatives and why they work. But "no student in math should use techniques if they don’t understand how it works" is just ridiculous. We constantly ask people to learn rules they can't prove. I'm not going to insist a first grader read *Principia Mathematica*.
We were discussing the l’Hôpital rule and why some teachers don’t approve of its usage. And I am talking about learning math for high school students and about my philosophy for my college students. I am not talking about teaching pedagogy for grade school.
> If you cannot prove it, you cannot use it What about the Implicit Function Theorem? Did you prove it before using it?
You don’t need the implicit function theorem to prove l’Hôpital’s rule. They need to at least appreciate the graphical sketch using the mean value theorem. Math is not learnt properly without proofs and not all proofs at every level need to be rigorous but a proof is required.
At the college level, yes, I prove that theorem for my class. In any case, that theorem is not needed at lower levels and any derivatives (like ln x etc.) can be proved using implicit differentiation.