i remember i poof it myself in the senior year in high school(in China, high school only has three years' courses), i was so happy. the profound understanding in calculus really launched my physics scores. and now i'm a undergrad in physics.
This was my first real crack at real analysis. Although I definitely didn't get all of it, I am starting to see some of the relationships analysis cares about (at least in this proof), and it makes me quite excited to study it!
Great video! In class, my teacher sketched out a non-rigorous proof of the 0/0 case where you use the local linearizations of f and g to approximate the ratio. Much more intuitive, but not as ironclad.
At 4:17 I cannot unsee WTF as another anachronym. I mean delta, WTF?! Also 6:11 now reveals he is not making this video from his home as I earlier suspected.
At my university, there is a severe lack of space to work alone. People are constantly looking for empty rooms to study, and 6:08 gave me flashbacks of this past semester.
My accidental way of doing it was: 1: trying to prove cauchy's theorem as a personal exercise, i wanted to graph it 2: find the relative max and mins of f'(x)/g'(x), let this be h(x), we differentiate h(x), and let x be 0, we have 0=(f''(x)g'(x)-f'(x)g''(x))/g'(x)² 3: we simplify and get f'(x)/g(x)=f''(x)/g''(x) when h'(x)=0, same thing but we have it differentiated once more, we just remove it for l'hopital's rule 4: the not rigorous part, add lim in both sides :p
Can we just expand f and g with power series and we know that the constant terms are both 0s at the limit. Now given the ratio of f’ and g’ is L, meaning that the ratio of their 1st order term at the limit is L, now we integrate numerator and denominator and add back the constant term, the ratio is (0+ a*L+O1)/(0+a+O1) where a is just the result of integration. O1 is small. The ratio is still L at the limit.
Thanks a lot for showing us this proof. I have two questions Could anyone write me the real analysis book's name he mentioned I couldn't catch the author's name (my english is inadequate for that:) )? How do we know that by which constant should we multiply epsilon why and WHERE does he know g(X)^2 / 4[f(x)+g(x)] I saw different proofs and epsilon was multiplied by different things how can we decide/consider that factors?? thanks..
I wonder if you set c=0 during integration, aka from nding the principal integral of functions, that this works in reverse? Int(f)/int(g) approaches the same thing as f/g, and this could be use to define the behaviors of non-integratable functions
Dr peyam,I have a question......is it valid proof if I use cauchy mean value theorem to prove l hopital rule......as I am struggling quite a bit with epsilon delta proofs..... I am just a high school student so if you have any suggestions or explanations plz make it extremely simple.....thank you very much
Greetings from Belgium! A video suggestion : proof of Mertens theorem, i.e. the (Cauchy) product of a convergent series and an absolutely convergent series is a convergent series 😊
If you have that f/g infinity/infinity you cant rewrite that as 1/g/1/f as the t gies to 0 and have again 0/0 so repeta the seame proof for 1/g and 1/f?
3:55 I had to do a double take as "wait if g can never equal zero then how is g(x) going to zero" before realizing you meant the function of g couldn't be zero, g(x) not zero, but g can still be zero
@@drpeyam Grant Anderson from 3blue1brown, in his essence of calculus video on limits, made a really good intuitive proof on this which explains why this is true
You could argue about rigor but i think it also works if you define u(x)=1/f(x) and v(x)=1/g(x) in a neighborhood of a (the zero) so when f(x) and g(x) go to zero then u(x) and v(x) go to infinity. At that point you may use hopital on u and v or f and g, depending on what you need.
Case 0/0 is quite easy or I am wrong Let f(a) = 0 and g(a) = 0 it should be true for continuous functions Let's write our limit as follows limit((f(x)-f(a))/(g(x)-g(a)),x=a) Let's divide numerator and denominator by x-a limit(((f(x) - f(a))/(x-a))/((g(x) - g(a))/(x-a)),x=a) Let's use limit arithmetics to get limit(((f(x) - f(a))/(x-a))/((g(x) - g(a))/(x-a)),x=a)=limit((f(x)-f(a))/(x-a),x=a)/limit((g(x)-g(a))/(x-a),x=a) and we have limit of the backward difference quotient Case infinity/infinity is more complicated for me
they are arbitrary real numbers you choose so that you can prove the limit exists. If the difference |L-F(x)| < epsilon when |x-a|< delta that means you can approach L as much as you wish provided you approach a. No matter how small epsilon is, 0.000000001 or 0.00000000000000000000000000000000000000000000000000000000001, you'll always be able to get closer and closer to L. Brillian idea isn't it?
@@drpeyam why should we use the "modern" version of his name? The marquis signed his writings with Hospital not Hopital. Over the years the s in some words has been supplanted by accents in the next vowel in this case ô. this change of names, however, occurred after the death of the Marquis, so his name should remain unchanged. Hospital is more correct. Among other things, the theorem was thought by bernoulli.
@6:08 Hmmmm room 420 hahaha
Why is 420 funny?
Nice.
Actually when he was teaching, I'd remember you, @blackpendrepen.. coz his tune of voice match with you😂😂😂 really...
Haha bprp knows le funi number
@@gregoriousmaths266 weeeeeeed
We should really call it "Bernoulli's Rule, sponsored by Guillaume de l'Hôpital".
Haha
Sorry I was looking for the conference. I guess I should go to room 420 now. Thanks
🤣🤣🤣
@ 6:12 "are you practicing a talk?" ,"no just my fields medal acceptance speech"😂😂😂
Hahaha 😂😂😂
Your enthusiasm is contagious, Doctor. Great video!
I really really like L'Hopitals rule! It's usage is very easy and clear but the proof itself has many cases you have to account for :)
My new life motto:
@4:52 Epsilon means happiness
Love the excitement and enthusiasm shown here! Keep it up Dr P.
at first it was weird then it was hilarious and honestly best teacher ever
Your enthusiasm suits the beauty of the proof.
i remember i poof it myself in the senior year in high school(in China, high school only has three years' courses), i was so happy. the profound understanding in calculus really launched my physics scores. and now i'm a undergrad in physics.
Wow, you proved it in high school!
i love how there was a mini proof in this big proof. this has to be one of my favorite proofs ever
What an adventure of mathematics, an extremely good proof with complicated triangle inequality.
When you wrote "want to find" as "wtf" I most certainly did not read it as "want to find".
This was my first real crack at real analysis. Although I definitely didn't get all of it, I am starting to see some of the relationships analysis cares about (at least in this proof), and it makes me quite excited to study it!
Great video! In class, my teacher sketched out a non-rigorous proof of the 0/0 case where you use the local linearizations of f and g to approximate the ratio. Much more intuitive, but not as ironclad.
I just love that you are so excited to show us the proof
Awesome! I used to hate Analysis, but now I'm beginning to like it
GREATEST TEACHER AND SHARER EVER ！
Once again, a great vid by Dr. P. 👍
22:48 That's the best math channel ever. no contest.
Wish I had a professor like him. Professor at my college seem like they hate teaching.
At 4:17 I cannot unsee WTF as another anachronym. I mean delta, WTF?! Also 6:11 now reveals he is not making this video from his home as I earlier suspected.
I was using this thm since i was 16 and now understanding the proof in 24.
you're amazing peyam joon, thank u
Bernoulli's rule or l'Hospital's rule ; I forgot the proof procedure of this rule. So, pleased.
"Epsilon means happiness"
Phrase to remember..😂🙂
At my university, there is a severe lack of space to work alone. People are constantly looking for empty rooms to study, and 6:08 gave me flashbacks of this past semester.
It's basically an application of the general (extended) mean value theorem for derivatives.
the textbook by Pugh omits too many steps😭😭in the process of part A, thank you very much!
My accidental way of doing it was:
1: trying to prove cauchy's theorem as a personal exercise, i wanted to graph it
2: find the relative max and mins of f'(x)/g'(x), let this be h(x), we differentiate h(x), and let x be 0, we have 0=(f''(x)g'(x)-f'(x)g''(x))/g'(x)²
3: we simplify and get f'(x)/g(x)=f''(x)/g''(x) when h'(x)=0,
same thing but we have it differentiated once more, we just remove it for l'hopital's rule
4: the not rigorous part, add lim in both sides :p
i can't put a sub-zero when i need to say a point instead of x, but you know where they have to go
Can we just expand f and g with power series and we know that the constant terms are both 0s at the limit. Now given the ratio of f’ and g’ is L, meaning that the ratio of their 1st order term at the limit is L, now we integrate numerator and denominator and add back the constant term, the ratio is (0+ a*L+O1)/(0+a+O1) where a is just the result of integration. O1 is small. The ratio is still L at the limit.
10:49 me when i learn knew calculus concepts such as the Fundamental Theorems of Calculus.
Thanks a lot for showing us this proof. I have two questions Could anyone write me the real analysis book's name he mentioned I couldn't catch the author's name (my english is inadequate for that:) )? How do we know that by which constant should we multiply epsilon why and WHERE does he know g(X)^2 / 4[f(x)+g(x)]
I saw different proofs and epsilon was multiplied by different things how can we decide/consider that factors?? thanks..
Real analysis by Pugh
@@drpeyam Hellooo thanks a lot for your respond that makes me really happy .Finally I found the book😃😃 Greetings from Türkiye....
Beautiful proof, greetings from Mexico.
The guy brings his A game every time !!!
This is beauty of mathematics it makes people innocent ,like him
Thanks!
I wonder if you set c=0 during integration, aka from nding the principal integral of functions, that this works in reverse? Int(f)/int(g) approaches the same thing as f/g, and this could be use to define the behaviors of non-integratable functions
Ummm nice question man, I think it's not true, but can't give you a counterexample 🤣
It’s actually my first time seeing WTF=want to find. Math is exciting.
I am a high school student, please tell me what is epsilon??? I was just taught lhopital without any proof!!!!
What book were you referencing at 14:55 ? Cant seem to find it online.
Real Analysis by Pugh
@@drpeyam Thank you very much! I'd never guess that name.
2 years later, this thread saved me a lot of time! thanks
YALL HE SPEAKS FRENCH??? so dyou remember when i said best teacher ever? yeahh
Dr peyam,I have a question......is it valid proof if I use cauchy mean value theorem to prove l hopital rule......as I am struggling quite a bit with epsilon delta proofs.....
I am just a high school student so if you have any suggestions or explanations plz make it extremely simple.....thank you very much
If f(x) and g(x) are differentiable and f(a)=g(a)=0, then lim(x->a)f'(x)/g'(x)=lim(x->a)[f(x)-f(a)]/[g(x)-g(a)]=lim(x->a)f(x)/g(x) .
Hey do you have a video explaining the logic at the start , about epsilon and delta ? It would be great
Yes google epsilon delta dr peyam
Greetings from Belgium! A video suggestion : proof of Mertens theorem, i.e. the (Cauchy) product of a convergent series and an absolutely convergent series is a convergent series 😊
If you have that f/g infinity/infinity you cant rewrite that as 1/g/1/f as the t gies to 0 and have again 0/0 so repeta the seame proof for 1/g and 1/f?
This video was sponsored by Bernoulli
Dr peyam Is it valid to proof this using cauchy's mean value theorem?
I think you forgot an absolute value at 20:55 when stating the reverse triangle inequality! However it does not have impact on the proof!
Since |a-b| = |b-a|, you can write the inequality that way without it being weaker.
Doesn't t need to go to zero, in order for your substitution for f'(x) and g'(x) early on to have worked?
4:20 “WTF”
420 time stamp xd
Great video! Thank you sir !
Hello. Thanks. Always excellent.
Merci 😄
3:55 I had to do a double take as "wait if g can never equal zero then how is g(x) going to zero" before realizing you meant the function of g couldn't be zero, g(x) not zero, but g can still be zero
this is the second comment. Hello, Dr. Peyam !
Hi 😄
@@drpeyam Grant Anderson from 3blue1brown, in his essence of calculus video on limits, made a really good intuitive proof on this which explains why this is true
@3blue1brown
Great video!
Love ya man ♡
Nice Video
I wonder how people come up with such proofs. Even considering the fact that no technology was used back then
Now I would like you to do the same proof again, but this time, WITHOUT any note and Internet, with only papers and pens wwwww
So if i skimmed your video correctly you have proven the special case of L’hôpital’s rule (“0/0”) but how about (infinity/infinity)?
Basically the infinity/infinity proof is similar, except you basically do things at infinity except at 0.
Dr. Peyam's Show ty ;)
You could argue about rigor but i think it also works if you define u(x)=1/f(x) and v(x)=1/g(x) in a neighborhood of a (the zero) so when f(x) and g(x) go to zero then u(x) and v(x) go to infinity. At that point you may use hopital on u and v or f and g, depending on what you need.
Dr Peyam why did you switch from doing videos on a chalkboard to whiteboard? :O
My clothes got messy! And whiteboards are cleaner and I can use colors on them
aka proofing the gangs are right
4:05 I think we only need g’=/=0 in the proof
Case 0/0 is quite easy or I am wrong
Let f(a) = 0 and g(a) = 0 it should be true for continuous functions
Let's write our limit as follows
limit((f(x)-f(a))/(g(x)-g(a)),x=a)
Let's divide numerator and denominator by x-a
limit(((f(x) - f(a))/(x-a))/((g(x) - g(a))/(x-a)),x=a)
Let's use limit arithmetics to get
limit(((f(x) - f(a))/(x-a))/((g(x) - g(a))/(x-a)),x=a)=limit((f(x)-f(a))/(x-a),x=a)/limit((g(x)-g(a))/(x-a),x=a)
and we have limit of the backward difference quotient
Case infinity/infinity is more complicated for me
Want to find 420
lmao when he uses wtf for want to find
which analysis book you said . Can anyone write it
Ross
Pugh
One question, what are epsilon and delta?
they are arbitrary real numbers you choose so that you can prove the limit exists. If the difference |L-F(x)| < epsilon when |x-a|< delta that means you can approach L as much as you wish provided you approach a. No matter how small epsilon is, 0.000000001 or 0.00000000000000000000000000000000000000000000000000000000001, you'll always be able to get closer and closer to L. Brillian idea isn't it?
Nice job
Isn't it L'Hospital?
Hôpital is the correct way to write it, unless you’re speaking 1800s French
@@drpeyam Okay, thank you. Have seen many people spelling it with an s.
@@ZonkoKongo that's what the ^ is for.
@@drpeyam why should we use the "modern" version of his name?
The marquis signed his writings with Hospital not Hopital. Over the years the s in some words has been supplanted by accents in the next vowel in this case ô. this change of names, however, occurred after the death of the Marquis, so his name should remain unchanged.
Hospital is more correct. Among other things, the theorem was thought by bernoulli.
it is an alternate spelling, but then again this rule is so sick it makes sense
Avevana pensato che forse parlavo con se stesso, ecco perche sono entrati nella classe ed hanno trovato quella scusa! 😀😉
If you divide f(x) by x, you have f'(x), x is cancelled.
do you follow a script ?
❤
Lmao I loved 6:08
Also I don't think I've ever noticed you're left handed! Unless you're ambidextrous
I know your proof is mathematically rigorous. But there is a much simpler proof at th-cam.com/video/AiQUe8M8dj8/w-d-xo.html
are you INDAIN...
No
Dude sounds effeminate.