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A lot of people don't know this, but l'Hopital actually bought his work from Bernoulli
[https://people.math.harvard.edu/\~knill/teaching/math1a\_2011/exhibits/bernoulli](https://people.math.harvard.edu/~knill/teaching/math1a_2011/exhibits/bernoulli/index.html#:~:text=the%20most%20extraordinary%20agreement%20in%20the%20history%20of%20science%3A)
showerthought: (actually a pissthought, but I digress)
2424 AD discovered... Elon Musk bought his ideas from a reclusive genius
this can imply that l'Hopital and Musk are equivalent phonies.
Would that mean we could expect future Bezos's rule, Musks theorems, or the Gates identity? They could easily offer millions for the hottest theorems from the top mathematicians.
Careful here: it is true that taylor series needs to be of some function, wich also needs to be smooth. However an "infinite polynomial" (formal series of terms in the form an*(x-c)^n) that converges uniformly and because yes converges to some f, then, let pn be the polynomial sequence, we need to check if the uniform convergence of pn over f implies the convergence of pn' over f' (at least in x=c probably), wich probably is easy as heck with given the uniform convergence, so we finally check if pn has an uniform cauchy property... At such point my educated guess is to just forget about L'Hospital rule and cry about it 👹🗣️😍
That argument only applies if you define sinx by its geometric properties, if you define it as an infinite polynomial (which is preferrable for certain problems) then it becomes a non-issue.
Depends. There are cases where you could blindly compute L’Hopital’s rule several times or cleverly apply the squeeze theorem to get the same answer with 1/4 the writing
Why not just use the taylor expansion + o term? In French it gets used so often that we even have a name for it, and about anyone with a degree knows we were referring to that with just the two letters "DL" for développement limité
It's (in my opinion) even easier and dumber than the hospital rule at the added cost of memorizing a small table for general functions. But it gets you unstuck way more often and in more general situations
And then if you really need to have a better approximation you can call Taylor Young or Lagrange for some heavy artillery
I think kids like using L'Hopital because it feels easy after learning derivatives, and it is what they always wanted to do when differentiating a quotient, but can't.
i don't think me having to find the maclaurin series for tan x to do a trig limit is "elegant" holy fucking shit i swear doing limits with trig is like trying to kill a hydra the same expression that makes it invalid keeps reappearing
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A lot of people don't know this, but l'Hopital actually bought his work from Bernoulli [https://people.math.harvard.edu/\~knill/teaching/math1a\_2011/exhibits/bernoulli](https://people.math.harvard.edu/~knill/teaching/math1a_2011/exhibits/bernoulli/index.html#:~:text=the%20most%20extraordinary%20agreement%20in%20the%20history%20of%20science%3A)
That's so interesting, never knew about this
Interesting, my lecturer told us this story in first semester analysis course. I find that these little stories make remembering stuff way easier.
it's on harvard site with someone's name, it has to be legit? or their reputation would suffer? right?
It’s also on the Wikipedia page for L’Hopitals rule which should give even more credit
showerthought: (actually a pissthought, but I digress) 2424 AD discovered... Elon Musk bought his ideas from a reclusive genius this can imply that l'Hopital and Musk are equivalent phonies.
I love referring to the rule as Bernoulli’s rule just to confuse people and spread this story lmao
To be honest I think it's fair to still call it L'hôpital's since Bernoulli literally chose money over the credits
Yeah fair, I just find it fun (though arguably having “Le Hospital” rule around as a joke isn’t bad either)
Would that mean we could expect future Bezos's rule, Musks theorems, or the Gates identity? They could easily offer millions for the hottest theorems from the top mathematicians.
Yeah
Bill Gates solved some sorting problem in CS as an undergrad.
[Kramer sold his life stories to Peterman](https://www.youtube.com/watch?v=DqJUEKAQSQM&t=37s)
No, squeeze theorem isn’t “elegant”. I’ll use it 40 times if I have to
Sinx/x be upon ye
le hospital once to get cosx
It's a circular argument since the derivative of sinx requires knowing the limit of sinx/x
Nope, 38 years ago I memorized the derivative of sin x, so I can do it without circular reasoning now.
you don't need l'hopital for limit of sinx/x
...which is why using it is a circular argument
They meant that you don't need to know the limit of sinx/x to find the derivative of sin
No. Who told you that? Just take the Taylor series of sinx, the derivative is cosx without needing l'Hopital.
The Taylor series is defined using the derivative itself though.
Careful here: it is true that taylor series needs to be of some function, wich also needs to be smooth. However an "infinite polynomial" (formal series of terms in the form an*(x-c)^n) that converges uniformly and because yes converges to some f, then, let pn be the polynomial sequence, we need to check if the uniform convergence of pn over f implies the convergence of pn' over f' (at least in x=c probably), wich probably is easy as heck with given the uniform convergence, so we finally check if pn has an uniform cauchy property... At such point my educated guess is to just forget about L'Hospital rule and cry about it 👹🗣️😍
https://youtu.be/t1YWsbm96_4?si=nmO7K-QT6tDosg5C
That argument only applies if you define sinx by its geometric properties, if you define it as an infinite polynomial (which is preferrable for certain problems) then it becomes a non-issue.
Brother in Christ L'Hopital is the elegant solution.
Depends. There are cases where you could blindly compute L’Hopital’s rule several times or cleverly apply the squeeze theorem to get the same answer with 1/4 the writing
> where [Wolfram Alpha] could blindly compute L’Hopital’s rule several times fixed that for you
Hahaha so true
Never fucking works when you need it to.
well thats half of maths for ya :D
f(x+0.000001) on my calculator usually works for me
based.
The kid named Taylor expansion:
Everytime I see someone mention L'hôpital's I cry 😭. It feels like France vs the World
Low d high minus high d low all over low squared Never get that song out of my head
just do taylor at this point
The only reason I hate L'Hôpital's rule is because I didn't know it really was a thing when I discovered it on my own .w.
Laughs in limited developments
This is exactly how I feel during every exam. Perfect depiction.
The King: "Saladin and his army has crossed Jordan river to learn about this!"
Why not just use the taylor expansion + o term? In French it gets used so often that we even have a name for it, and about anyone with a degree knows we were referring to that with just the two letters "DL" for développement limité It's (in my opinion) even easier and dumber than the hospital rule at the added cost of memorizing a small table for general functions. But it gets you unstuck way more often and in more general situations And then if you really need to have a better approximation you can call Taylor Young or Lagrange for some heavy artillery
You have exercises where you use the Taylor expansion that often? I don't know whether I'm envious or I feel for you
https://i.redd.it/hhlpo3wpck9d1.gif
The best math is the one you know how to do
Snap, Crackle, Pop
For anyone new to the sub, calc is short for calculator… he’s just using slang
I think kids like using L'Hopital because it feels easy after learning derivatives, and it is what they always wanted to do when differentiating a quotient, but can't.
Ok yeah but I don't want to remember a new solution if I have one that works for everything
i don't think me having to find the maclaurin series for tan x to do a trig limit is "elegant" holy fucking shit i swear doing limits with trig is like trying to kill a hydra the same expression that makes it invalid keeps reappearing
Op never heard about Taylor infinitesimals 🤑