L'Hopital's Rule for the 0/0 Indeterminate Form (proof)

[@cromthor]

The "^" over the "o" usually means, in French, that an "s" was dropped. So it's either "L'Hospital" ("s" and no "^") or "L'Hôpital" ("^" and no "s"). Never both: "L'Hôspital" is wrong. But either of the other ways is correct. I'll go for... mmm... The one with the "s", more old-fashionned. And because "hôpital" is the current french word for... "hospital". I guess if I spoke English, I would go for the "ô" ;)

[@blackpenredpen]

Oh wow! Thanks for sharing!! I have been wondering that for a long time!

[@PackSciences]

Guillaume de l'Hospital/Hôpital prefered using Hospital for printed documents, as circumflex accents weren't easy to manipulate at the time. In Analyse des Infiniments Petits, the book that uses the named rule, his name his spelled l'Hospital, but it's only in the reeditions of the book, as the original handwritten didn't have any author note. However, when he was writing handwritten, it is said (from what I've heard in class) that he wrote it l'Hôpital. Anyways, now the use of "accents are difficult to print" doesn't make sense anymore, but if you have a non-AZERTY keyboard, it can be pretty handy to spell it Hospital. Therefore, I recommend using Hospital for quick documents, and Hôpital for long documents and handwritten, as it was taught to me. However, you can find examples of both in the literature so I guess the choice doesn't matter that much. Alternatively, in the Encyclopedia of Diderot and D'Alembert, his name is spelled "Hopital" (probably to avoid printing ô character).

[@cromthor]

You're welcome, and btw, I like it how your enthusiasm and love for math show in your videos. "So cool, isn't it?" :D I feel the same -and that makes my highschool students laugh ;)

[@gnikola2013]

cromthor This is amazing

[@gnikola2013]

PackSciences that's cool!!

[@TheBlueboyRuhan]

Mate, every person in my A level maths class I recommend and I love your vdeos to the point I want to do maths at university You're an absolute youtube star and I hope you keep making the great, consistent content :)

[@matthewstevens340]

Sir Rahmed if only my friends liked maths enough to do the same 😁

[@idontknow1630]

so you doing maths at uni then

[@arkabanerjee1091]

Sir thank you so much. Till now I used L'Hospital rule countless no. of times for evaluating scary looking limits with ease, but I didn't know why this method worked in the first place. Love maths and love you 3000 ❤️.

[@shacharh5470]

It's nice to see you doing proofs on your channel, you usually do more technical exercises.

[@blackpenredpen]

Yay. I do some proofs here and there. : )

[@craftbuzzwonky4752]

@@blackpenredpen And I am pleased at your proofs!

[@ThePron8]

just a quick question: why is this theorem so much hated? My professor of calculus in high school warned us from using it and at university my professor of calculus also said it. I can't understand, i find it super useful for evaluating limits of real function in one variable

[@blackpenredpen]

I think they mean the cases that don't use L'H rule for trying to prove derivative. For example, lim as x goes to 0, of sin(x)/x, should NOT be done by L'H. Since that's a crucial limit for getting the derivative of sin(x). And if we would going to use L'H on that , we would get cos(x)/1, but this would be what we called "circular reasoning"

[@IronLotus15]

I guess it is good though to have that sanity check, of checking that limit of sin(x)/x going to 0 is 1 by L'H.

[@ThePron8]

In general I felt a sense of rejection of this rule on their part. They encouraged us to use Taylor/MacLaurin series instead and their point was :" If you try to list the classes of real function in one variable who became easier to work with when you differentiate them, you will find out that they are only polynomials and trigonometric function with a simple argument".

[@52flyingbicycles]

blackpenredpen CIRCULAR reasoning with SIN and COS? I see what you did there.

[@blackpenredpen]

VeryEvilPettingZoo Excellent explanation and examples!!! LOVE IT!

[@alkankondo89]

Two coincidences involving this video: (1) I was just thinking about how I've seen more than one unrelated instances of people pronouncing it "L'Hôspital" when, in my experience, I've never actually seen it spelled that way. And (2), I was SERIOUSLY JUST TALKING WITH SOMEONE about this rule about 20 minutes ago, trying to justify why it's true but forgot how to prove it. Then, lo and behold, here is the video I need!

[@blackpenredpen]

Hehehe, I read your mind!! The vid was secretly for you!

[@Nobody-Nowhere-Nothing]

If you want to justify why it's true to someone then tell them this: If we are trying to get the limit x->a of f(x)/g(x), then just analyze what happens to functions separately as a x->a. Imagine zooming in at the point x=a for the function f(x), f(x) will start to look like a straight line. The more you zoom in, the more f(x) will look like it's derivative at the point x=a. Same thing goes for g(x).

[@davidseed2939]

Useful to show graphs of both functions wgich demonstrates L'H. Could also show power series expansion and see how that also fits. Also try seeing the relationship between sinx/2x and sin^2x/4x^2. Second case requires 2 uses of L'H graphing those functions also gives insight on the repeated appl8cation of L'H

[@gchow9665]

I learnt the same proof for L'Hopitol rule in my A-Level course, and this is baby proof!!!

[@PackSciences]

Note about the thumbnail: I don't think zeros are happy

[@blackpenredpen]

Oh come on Pack! Look at their smiles! : )

[@PackSciences]

Addition? Ineffective? Multiplication? You end up looking at a glass. The zero-life is inherently sad.

[@noname_whatsoever]

PackSciences Don't forget the restraining order denominators filed against them. : (

[@dougrife8827]

Here is a much simpler and more intuitive proof of the rule. One way to approximate the limit of f(x)/g(x) as x approaches c is to try a value for x that is very close to c such as deltax+c where deltax is a very small number. In most cases, you can use a calculator to approximate the limit by selecting a very small number for deltax say 0.00001 and calculating f(c+deltax)/g(c+deltax). Before modern calculators there was a way to simplify this calculation using calculus: Notice that in this indeterminate case f(c+deltax) can be approximated by deltax*f’(c) and g(c+deltax) can be approximated by deltax*g’(c) with exact equality as deltax approaches zero. So the limit can be approximated by deltax*f’(c)/deltax*g’(c) and approaches the original limit as deltax approaches zero. Notice that when we calculate this ratio deltax cancels out and we are left with just f’(c)/g’(c). End of proof.

[@ananyapathak8701]

Hey! Blackpenredpenbluepen?

[@blackpenredpen]

: )

[@lalitverma5818]

H

[@ananyapathak8701]

#YAY

[@lalitverma5818]

Deepa Pathak oh same country

[@ananyapathak8701]

Lalit Verma i am from mathematiconesia

[@haakoflo]

My mental image for l'Hôpital's rule was always a Taylor series expansion over and under the divisor, centered on a. Then find the lowest order term that is nonzero either over or under the divisor. Every higher order term goes to zero as (x-a) goes to zero.

[@anikethdesai]

I had a very intuitive proof for this. If say lim f(x)/g(x) x→a is giving me 0/0 or ♾️/♾️ then, it means that both f and g are reaching 0 or ♾️ as x→a together since f(a) = g(a) =0. If both f and g are reaching together, then it means that the curves of f and g start to coincide as x→a from both Left and Right Side. If both the curves start to coincide as x→a, then it means that the slopes or the derivatives of both f and g must be equal or slope of one (either f or g) must be a multiple of the other. Hence, L'Hopital rule works

[@splodinatekabloominate846]

The correct way is "L'Hôpital's Rule", because the accent on the "o" replaces the s"" after it. For example, "forest" in French is spelled "forêt", and the accent on the e replaces the s. Also, the namesake's name was spelled "l'Hôpital." Spelling it with the "s" would be like spelling "L'Hosspital"

[@Broody90]

I have no clue if it works as a proof, but a function f can be approximated as f'(a)(x-a)+f(a), also g(x) can be approximated as g'(a)(x-a)+g(a). if you plage it in lim f(x)/g(x) then lim [f'(a)(x-a)+f(a)]/[g'(a)(x-a)+g(a)]. If (third condition) f(a)=g(a)=0, that's means lim [f'(a)(x-a)]/[g'(a)(x-a)] = lim f'(a)/g(a) = f'(a)/g'(a)

[@narrotibi]

It's my most pleasant daily routine: watching the newest bprp video.

[@bradycall1889]

Now the medical hospital’s rule starts to make sense (at least in some cases)

[@bilz0r]

2:12 "Here is....." THE DEAL!

[@lunjapaobaite4071]

Please, do for Rolle's theorem slso

[@joebrinson5040]

I always enjoy your videos.

[@Koisheep]

0:30 amen to that. It was the part I hated the most from Calc 2... L'Hôpital's proof Ok after seeing the video... i swear I don't remember it being so simple? I can't find the point where I can confidently say, "this proof doesn't work for other cases" except for a -> ∞. And even then, You can make go to the u world and say let u=1/x. I wish I had known this channel in high school (((sad)))

[@ethan_martin]

I prefer L'H xD

[@blackpenredpen]

ME TOO!!!!!!!!!!!!!!

[@ugursoydan8187]

GOD BLESS YOU MY BROTHER

[@csanadtemesvari9251]

we used to do it with mean value theorem

[@manishKumar-mn9bt]

Thanks a very lot bro to clear the co nfusion

[@aidangarner1181]

Is the thumbnail loss?

[@BilalAhmed-on4kd]

what if g'(a)=0 ?, does this mesn that we can't use it again if after the first time we differentiate we still get 0/0?

[@rhaq426]

Can't this qualify as a general proof? Since l'hopitals rule only qualifies for indeterminate forms (0/0 and infinity/ infinity) if you've proven it for 0/0 you've infinity as well. All indeterminate forms can be written as 0/0. For example, consider a limit lim_x->c f(x)/g(x) which is infinity over infinity, one can equivalently say lim_x->10^+ (1/g(x))/(1/f(x)) which is 0/0 and we've proven l'hopitals rule for 0/0

[@siddharthpandey8516]

pretty easy proof. got to it myself. so intuitive :D

[@lalitverma5818]

Good proof

[@Amine-gz7gq]

I like this guy. best explanation !

[@ChefSalad]

You should do a fake L'Hospital's Rule question where you find functions f(x) and g(x): d/dx[f(x)/g(x)]=[df/dx]/[dg/dx]. I did this but couldn't find any functions where it really looked clear what was going on.

[@gnikola2013]

I prefer H'Lôspital lol

[@lordyabo9910]

Can someone pls help. I thought the limit definition of derivative is lim as h----->0 ((f(x+h) -f(x)))/((h)). Why his definition of derivative is different? am I missing something?

[@Cloud88Skywalker]

Is it legit to divide by (x-a) inside a limit of x->a ?

[@neonet310]

Yes because it never really technically becomes zero. Practically lim x->a (x-a) behaves like zero (it adds/subtracts nothing and makes things multiplied by it near zero, so you would treat it as such) except in division, where you are allowed to divide by it (since it isnt _actually_ zero). this also applies to lim x->0 (x). 1+x=1 (approaches 1 since you add less and less), 1-x=1 (approaches 1 since you take less and less), 1·x=0 (approaches 0 since the decimal spaces go way to the right), and x/x=1 (since x isnt really ever 0 itself and is a valid division)

[@yge1035]

neonet 1™ but lim x->0 1/x is undfind... And if y=x-a And lim x->0 then y=0 So if you'll devide something by y it'll be undefind

[@yge1035]

Oh no because you devide both things by y then its ok

[@yge1035]

Or maybe not, I know that if you'll devide both sides by zero you just destroy the whole equation, but maybe if the limit is 0 its different

[@BasOomen]

Y GE that limit 1/x is actually undefined because you get different answers depending on whether you go from 0+ or 0-. Respectively they _approach_ +inf and -inf.

[@antoine2571]

learning proof by earth to be able to use it in class ahahahahah teacher will hate me

[@ghivarlynadiefaddarquthni5170]

How about proofing in other indetermined situation like infinite/infinite and etc?

[@hobodawg9364]

That was *SUPREME*

[@blackpenredpen]

As stated in the thumbnail!

[@Kumar-oe9jm]

This is so much better than Dr Peyam's

[@punditgi]

Many thanks for this video!

[@НикитаЧистяков-х8б]

I know it's too late to ask but... It's allowed to calculate such limits of infinity/infinty indeterminate form situation with L'Hospital rule,isn't it? But why does this method work in this case?

[@MT-od6by]

just take reciprocal of x as x goes to 0 and you get infinity.

[@etienneparcollet727]

L'Hôpital. Because the circumflex stand for an obsolete s.

[@dranxelaa6770]

Étienne Parcollet +1

[@IronLotus15]

Would L'Hospital be valid? Without the circumflex?

[@etienneparcollet727]

Yes considering he is not a recent mathematician. It is quite old-fashioned.

[@yuxuanlin7882]

wow,the most simplest proof

[@dhyeypatel5448]

Find Inverse of x + sin x?

[@csanadtemesvari9251]

since sin x is not bijectiv, I don't think it's invertable

[@arthurreitz9540]

Csanád Temesvári f(x)=x+sin(x) f'(x)=1+cos(x)>=0 It is bijectiv

[@csanadtemesvari9251]

i didn't go in that deeply, but since it's complicated, it must be true :)

[@rikkigunawan3326]

Can you help me integrate {1/((x^2)*(27x^2 + 6x - 1)^(1/2))}

[@dentionyoll3224]

What’s up with these thumbnails?

[@blackpenredpen]

Summer vacation!

[@andresxj1]

Why couldn't g'(a) be equal to 0 in the particular case that f'(a) is also equal to 0, so we could do L'Hôpital again?

[@blackpenredpen]

It's just a baby case this in vid. In reality, yes it could. And then you might be able to keep going with L'H.

[@boraidaa1313]

l'Hospital, the same way he himself spelled it! #yay

[@mariomasters1]

I thought the thumbnail was "Loss" lmao

[@Mau365PP]

I love baby proofs now 😀

[@blackpenredpen]

LOL!! = D

[@نعمللوحدة]

Love u 2 muchhhh

[@lianebans7885]

Amazing man

[@SteamPunkLV]

pf for proof? that's a new one :p

[@aneecraft2350]

HOSPITAL?

[@carultch]

The s is silent. It's usually written with either a hat over the o and no s, or with the s. French spelling makes no sense.

[@Flanlaina]

Luo Pi Tai Lu?

[@naturemeets]

the BEST !!!! man.

[@flowerwithamachinegun2692]

Neither my teacher nor my classmates would understand that :))

[@john-athancrow4169]

I didn't know how to say it. Sorry.

[@john-athancrow4169]

I thought f instead of g

[@Berkolika]

thank youu

[@xaxuser5033]

Why u don t choose me yeeeeeeeeh! #yay

[@themeeman]

I prefer Bernoulli's rule

[@ivanenkoandriy8753]

Is it legal do divide top and bottom by (x-a)? It is approaching to 0, so we divide by zero and result in undefined...

[@aadityajha7502]

Lim of (x-a) is 0 directly implies it is not zero . This is the difference between approach something

[@MohitBaboria]

It is not exactly zero Even in the definition of derivative The denominator h approaches to zero And we somehow cancel the h using algebric methods while differentiating any function

[@lorentzisagoat8409]

In France the Hospital rule is forbidden even though it was invented by a Frenchman 😭

[@carultch]

Do they require you to call it Bernoulli's rule?

[@rain_357]

Thanksss.

[@opufy]

This guy should be making like 500k USD a year working for tesla or what have you.

[@blackpenredpen]

I am definitely not that good for Tesla but thank you.

[@opufy]

@@blackpenredpen you are, and you have a great personality and they'd want you after knowing you made this channel :)

[@shukradityabose7131]

i prefer whatever u prefer #YAY#BPRP

[@blackpenredpen]

Shukraditya Bose I prefer L'H

[@shukradityabose7131]

I prefer L'H then XD

[@blackpenredpen]

deal!

[@shukradityabose7131]

DEAL!#YAY

[@inconnu8267]

Perfect😄

[@l1nk353]

I SUBBED

[@SK-gy3ec]

Luv u sir

[@joshmcdouglas1720]

Lol you could’ve just used L’Hospitals rule to prove it on the first step lol

[@tahirrazakhan6507]

#YAA

[@soumyachandrakar9100]

L'hôpital rule is better

[@Absilicon]

L'hopital #YAY

[@john-athancrow4169]

Yes. I know it doesn't do anything with the quotient rule.

[@reeshav4955]

L hopital ftw

[@john-athancrow4169]

I thought d'able was enable but… no. Sorry.

[@majkgmajkg2613]

lots of comments, lots of compliments... can I (as your > 1y subscriber) just #YAY?

[@blackpenredpen]

Yes!!!!

[@tanujabudhori]

❤️⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹••••• from NDIA🇮🇳🇮🇳🇮🇳🇮🇳

[@MrBenten867]

Doremon

[@tuna3875]

I prefer YOOOOOLER

[@Pur_Jun]

#yay#yay

[@PhasmidTutorials]

:D

[@blackpenredpen]

yay!

[@Chronozia]

More math! #YAY

[@HARJINDERSINGH-wm4bn]

You are so hand some

[@emanueleusai10]

#YAY

[@deenaaalkotb]

again why you didn't answer my question x=e^x I will give you some hints first work with complex numbers and secondly work with laumbert w function if you don't know the solution say to me and I'll send it to you thanks

[@Jt_563]

Malayalies ndoo😂

[@15schaa]

I bet this comment will be forgotten. #YAY

[@eta3323]

Team L'Hôspital's!!! #YAY

[@mihailoopacic3756]

#YAY Abfoena hekrlmaek ❤

[@l1nk353]

#YAY FEATURE ME

[@blackpenredpen]

Next time!