Limits, L'Hôpital's rule, and epsilon delta definitions | Chapter 7, Essence of calculus
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- Опубліковано 17 чер 2024
- Formal derivatives, the epsilon-delta definition, and why L'Hôpital's rule works.
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Timestamps
0:00 - Intro
1:17 - Formal definition of derivatives
4:52 - Epsilon delta definition
9:53 - L'Hôpital's rule
17:17 - Outro
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
Portuguese: rose ✨
Vietnamese: ngvutuan2811
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By the way, there is a piece of math, commonly called "non-standard analysis", which makes infinitesimals a rigorous notion, thereby avoiding the need to use limits. That is, in the real number system something like 0.000....(infinitely many 0's)...1 doesn't make sense, it's not an actual number. But the "hyperreal numbers" of non-standard analysis are constructed so as to include a number like this.
I have no problem with that system. I think it's great to invent new math and new number systems meant to rigorously capture a useful intuitive notion, although the construction of the hyperreal numbers requires some questionable usage of the axiom of choice. But I do think it's important to first learn about limits, and how mathematicians made sense out of calculus using the standard real number line without resorting to infinitesimals. It's not a matter of clinging to old systems, it's because limits help to gain a deeper appreciation for structure and character of the real numbers themselves, which in turn will help to understand any extension of those numbers.
I still don't get the issue with calling it "infinitesimals", isn't it exactly the samething?
I always have an issue with infinitesimals, I do not understand what it means. To me, the concept of limits seems more reasonable, more understandable, and more intuitive. Can someone care to explain to me, why some people think of derivatives as infinitesimals, and what does it mean to have infinitesimals instead of limits?
3Blue1Brown But what if you start teaching people calculus using the rigorous infinitesimal, won't the whole concept be more 'natural' to people then?
Is not an issue, but it demands some deeper theory about the algebraic properties of the infinitesimals. The epsilon-delta way is the easy way.
i find the notion of numbers "approaching" far more nebulous and unconfortable than infinites and infinitesimals..
i recently had an discussion with a mathy friend of mine, and after a few hours of staring at (for my eyes) convoluted equations (involving absolutes and "for all" statements) i eventually found it "kinda agreeable", but i could never reconstruct it..
with infinitesimals there is no hopping around between approximations and exact values, and no pretending that i could somehow shove all convergent series into it. less crying and headaches.
and i guess the "lim (x -> y, f (x))" can just be expressed as "st (f (y + ε))" and "st (f (y - ε))" with ε being infinitesimal and st the standard function. i don't see what else there is to gain from limits..
and while i have no idea of what the axiom of choice entails, i don't think any use of it is any more questionable than any other. :P
If anyone cares as to why the variable "h" is used in the definition of a derivative via the limit as h approaches zero: It's the same reason as to why "h" appears in quantum mechanics. A German mathematian used "h" to stand for "Hilfsvariable" (auxiliary variable) and everyone just ran with it. For clarity, these two occurrences of the letter are not in any way linked. They just happened for the same reason.
0.0
interesting
i thought it was the "height" for some reason xD
Which mathematician was it?
Seeing as it's the Planck constant I would guess the answer is Max Planck.
@@klobiforpresident2254 thank you.
@@klobiforpresident2254 but the planck constant is not a variable?
I did not know there was this much good in the world.
me too
O how beauteous mankind is! O brave new world, that has such people in it!
Yeah it's over-satisfying 😌
He is an angel among calculus students
@@musik350 Aldous Huxley
Literally Everyone: You can't teach calculus to grade-schoolers...
3Blue1Brown: Watch me.
I learned calculus when I was in tenth grade.
So did the rest of my peers.
I learned it in 7th grade
I learned it in 6th! booya!
Chanon Ross I learned it in 1st.
Ishwar Karthik I did it in 6-7th grade
In my experience as a tutor, I've seen that econ students often have a better concept of derivative than the physics students. It's easier to imagine marginal cost (what happens to my cost function when I make ONE EXTRA widget) than to imagine, say, how much extra distance my car travels after a tiny change in time. This fact surprised me.
Nai!!! To exo paratirisi kego
Haha what absolute nonesense. You're delusional
Interesting. Instantaneous velocity seems pretty intuitive to me.
I have 1 doubt,
I applied it to any random point, where the ratio is not 0/0 it was f(x)/g(x) = 3/5 at x=1, ( f = 2x^2 +1, g = 5x )
instead of just writing the value of of f and g as 3 and 5, I toook there limiting value at x = 1,
to find that, I found their individual slope at x = 1, and multiplies by dx, ( essentially what happened in here, you multiplies the slople at x=0 and multiplied with dx to get the final value)
however it did not work for this value,.. why?
what you depicted seems to be very general, something that I should be able to use at any point.
please answer !
@@full-metal_zero0683 You can only use L'Hôpital's rule when just plugging in the values gives you 0/0
God, this is my favourite series right now. First year engineering student and it's refreshing to hear everything from a more intuitive angle. I wish schools taught as well as you did Grant.
Blayke McGregor watch his linear algebra series too.
Rt
Tau > Pi
@@Taulussa Yes Tau=2Pi so Tau is greater than Pi. Obviously.
g r a n t
This is the first explanation of L'Hopital's that has made any sort of intuitive sense, even though I've proven it formally in Real Analysis.
Where can I find the proof?
@@MrBillonaire99 there is one on the english wikipedia page for L'Hopital's rule
@@MrBillonaire99 pretty easy to show it with simple calculus too. Just need to do some algebra with limits
xD
@@lukedavis6711 Not the 'full' version of the rule.
How should the future generations pronounce your name ?
L'Hopital : yes
L Hospital rule😂😂😂
Just say it as 'LH rule'
Can anyone please explain y it can't be used to find new derivative formulas? He said the reason in the video but I don't quite get it
@@samr9408 There's something else about it that confuses me.
Let's say we know that f is continuous and differentiable and that f' is again continuous with domain of definition D. Then we have for all x in D:
f'(x) = lim_(h->0) ( (f(x+h) - f(x))/ h )
Since f is continuous we have lim f(x +h) = f(lim x+h)= f(x). Therefore, what we have is of the form "0/0". We can use l'Hospital abstractly and plug in the derivatives of numerator and denominator. But since also f' is continuous, this gives
lim_(h->0) (f'(x+h) - f'(x)) / 1 = f'(lim x+ h) - f'(x) = f'(x) - f'(x) = 0.
Hence, f is contant everywhere in D.
Where did I mess up?
@@lonestarr1490 In the denominator, when you derive h with respect to x you get 0, not 1. This is where you made a mistake.
5:36
"imagine you have 0 cookies and you're dividing them among 0 friends. See? It makes no sense. Now cookie monster is sad that there's no cookies and you are sad because you have no friends"
L'Hôpital, what an ugly word! I have sympathy to people who have speak that language.
I think it’s a great word, sounds fancy
But if you have sin 0/0 cookies, you have 1 cookie! Congrats!
In this case, doesn't it make sense to think that each friend gets 0 cookies?
@@joaomatheus6222 No, the problem is that there are 0 people getting cookies and 0 cookies BUT the nonexistent cookies HAVE to be divided equally between the 0 people who exist. How do you give 0 cookies to 0 friends? The action can't exist, because it can't happen
Thanks to your videos I achieved a 100% in my AP Calculus course!
sad this series is FINITE :((((
no, real analysis still an active field of research...
Mi Les the youtube series
ever heard of PBS Infinite Series
eule franz it is fine, aight?
If it was infinite, we’d have infinite amount of videos to enjoy. But we would also never get to watch all of them, this way, we can see everything they make
Content like this makes me confident about the fact that the good part of the internet far outweighs the bad part.
I have read The Feynman Lectures on Physics. Waiting for The Sanderson Lectures on Mathematics. Grant Sanderson teaches really way, and what I LOVE about him is that he teaches visually, for the sake of learning and understanding, not just for the sake of covering a topic. Thanks a lot. Ur long-time fan, Grant Sanderson.
This is the video that has the most utility and is the most accessible. My 12 year old son watched it, understood it, and was profoundly more interested in math after seeing it. And he was already a mathy kid.
Well done sir.
I absolutely love this channel. it is very selfless of you to create such great content for learners across the globe. The animation, examples and script all reflect the amount of effort you put in to truly inject your passion and expertise in the video. Keep up with the great work!
Just look at his subscriber count. Even though his work is definitely amazing, it's quite far from selfless, since with that subscriber count, he can probably live off of his work. Not saying this is a bad thing, because he does deserve that, but if you can comfortably live off of your work, it's not really selfless, but more of a job.
@@mozesmarcus6786 You definitely aren't wrong, but arguably this particular job creates more total value in the world than many of the other equally or better paying jobs a smart dude like him could do. In that sense it's selfless of him to pick a career that creates such a positive externality.
@@eccentricity23 In any case, it's a win-win situation for everyone involved!
@@mozesmarcus6786 it's selfless anyway
With that skill he could be doing the double the money he makes rn not posting his videos for free, not spending valuable time making accurate and intuitive presentations etc. ...
Turn on the subtitle at 5:36. A little joke from our great teacher.
Exactly, I was looking for this. "Just ask siri" (when he talked about dividing by 0)
I saw that, and was about to comment it.
Why though?
Siri give you a funny explanation
Me:”hey Siri, what’s 0/0?”
Siri:”undefined”
I love how every once in a while, rewatching this series gives me different insights on calculus.
Exactly.
Finally!
My problem with your previous videos were, that they convey the right intuition for the general case, but leave out the dangerous edge cases!
Only in university, through infinite series and limits I began to apply these rules with confidence. It's great to be sure you don't break anything, and I think it is one of the most crucial things to ponder about. What can be safely ignored (and what cannot) for arbitrary choices of some value within a given (changing) range.
20+ years after trying to understand the teacher at high school I now finally understand the basics of how calculus works, which they never bothered explaining when I was younger. Thank you so much.
This one 19 mins video was worth of the 1 week of 40 mins classes that was given to me in my high school. This does put a smile on my face.
Excellent video, as always. Limits are *the* essence of Calculus. The single most important concept to learn in a Calculus course.
The only thing I'm "complaining" is: why didn't you actually write down the eps-del definition of limit? You had the ground work all laid down, you just needed to finish it off by giving the actual definition!!! :)
He was just introducing the epsilon delta definition for real analysis, defining that would be unnecessary and would confuse others too much.
It is an exercise left for the viewer ;)
He said in the video that the RA definition would be quite technical for an intro to calculus
@@levilikesstrawberrymilk8539 as can be easy seen by inspection in a exercise left for the reader
Facts.
I've trying to fully understand all this concepts for several years (+5), you know the feeling of joy that is close to tears? That's how I felt at 17:23.
Thanks for the videos, 引き続き頑張れ！
Watched this video a few years ago back in like 8th grade. Now that I'm in university and understand the epsilon-delta definition of limits looking back you explain everything really well. Can't find such high quality easy to understand maths content anywhere else on the internet.
Man, you have honestly become my favorite youtube channel. You have a way of explaining things that give me a "penny drop moment" almost every video.
I would buy a pi creature plushie!
Mageling55 Me too.
Mageling55 I want the one with the 'pause and ponder' pose or that pose when the brown pi gives a really confusing theory and the blue pis are like WTF
same
Me too, if the EU shipping was reasonable
+
An awesome video from an incredibly helpful and enlightening series! Watching your videos really fuels my passion to learn and understand. Your clear and thoughtful explanations, along with the fantastic animations, do such a great job of building a deeper understanding and help to expose the beauty of mathematics. Thank you!
Loved these series bro. As a phys graduate, it's refreshing to see someone explain such a "basic" and very formulaic concept in calculus and really define both physically and theoretically what a derivative actually means!
I ve been watching your videos since very a long time, but never saw the series in order and completely. I ve never been taught this much intuition for math neither in high school or at the university (and im studying bloody physics). I hope there are more series to come, this is really the best math content there is to find in youtube.
Every educator should show or suggest students to see these videos: You are awesome. Thank you.
Thank you for teaching me calculus in a way that makes sense to a 9th grader. This makes sooo much more sense than what parents tell me, so thank you!!!
9th grader, studying calculus, studying quantum mechanics and relativity and working on a theory of my own all thanks to this guy. I will owe my knowledge to him. Me and my partner are working on various projects. I love your vids and I support you. Thank you for educating me
4:44 That was a very beautiful detail with the opening and closing tag. It made me smile. Thanks for the great video!!
Please make a video on Laplace transform and Fourier transform...
+Abbas Rangwala
Fourier transforms now up!
ua-cam.com/video/spUNpyF58BY/v-deo.html
(no, I'm not 3Blue1Brown himself)
I believe there is some video about laplace transform from khan academy whose the person whose doing that is grant (3b1b)
Thanks, I highly recommend this video to my students. Fantastic!!!!!!!!!!!!!!!1
Comment #3:
Thank you very, very much.
I have not only understood what you were teaching - finally - but understood a fundamental problem in my earlier studies. You have moved me forward. I am deeply indebted.
David Lixenberg
wow, these animations and your vivid explaing really help me to not only get my brain to grasp that stuff but see the actual beauty in math. Thank you Sir.
i don't understand 80% of the videos but I can't stop watching them
He should make ASMR...
Cool. I never understood why L'Hopital's rule works. Not anymore!
It's not completely wrong bty since l' hopital is French word for the hospital
@@bhaskarpandey8586 LOL! hospital rule!
@@Anonymous-df8it cuz dis shit do be puttin me in da hospital
@@NewWesternFront ???
@@Anonymous-df8it ligma
Thank you very much for your French subtitles . I am learning French and these subtitles are a Big help . Also your wonderful graphics increase my understanding of complicated Math .
I love going back and re-watching ALL of your videos as refreshers, but I especially love this series.
16:15
3b1B: "Actually discovered by Johann Bernoulli, but L'Hôpital paid him for..."
Me: «TIRE SCREECH!!!»
Hahaha. Did you catch the Pythagorean triples on clay tablets centuries before Pythagoras?
@@williejohnson5172
Yes but that's different, isn't it? Because Pythagoras was the guy who proved it which is what really counts.
never clicked on a video so fast.
never been so happy to see a notification
I clicked in dt nanoseconds.
I just wanted to say thank you for all your teachings! You are a truly gifted teacher!
These classes are wonderful. Unhappily at the university I had lousy classes about these matters. Now I am starting to understand these matterd. Congratulations, dear Professor.
I thought the goals thing at the beginning of the video was a really good addition to the format of the video
I've somehow gone my whole life without having an intuitive understanding of L'Hopital's Rule. Now it makes perfect sense. Thanks!
Love it, these courses saved my life a lot, I keep watching those!
My comment on Chapter 6 has just been shown now for the first time at 2:27, the Greek delta. Great graphics and deep explanations in these videos and I love it!
parfait....toujours respectueux !!
Super video ! Thank you !
The production value and clarity of your explanations makes the abstract concepts in real analysis clear. :D Thanks for your videos
Love this series. With regard to , I usually try to tell my students to think of d_x_ as finite in size, and so small that (d_x_)^n can be safely thought of as zero for n > 1. I explain that it's a notation that abstracts away the limits by writing 'd's everywhere, with the expectation that ratios of 'd' quantities converge in the limit where higher order terms vanish.
I'm computationally biased, so I usually follow that up showing ratios of finite differences on a discrete grid, and extend that intuition to the real (or rational) numbers by visualising the convergent behaviour when between any two mesh points, there is another mesh point. But the visualisation of sampling on a mesh helps give the the students the intuition for why behaviour is linear in these limits.
NB: I usually teach 3rd year or higher undergraduate classes, where I am repairing poorly constructed intuitions the students may have.
Thank you very much Sir, you are an amazing lecturer , I wish I met a teacher like you when I was in high school/university, I wish you will never stop on producing more incredible videos. I recommended your videos to all my friends at the university. please carry on. btw, if it happens that you visit Japan one day, please let me know, I would like to invite you to
see the city I live in "Nagoya"
Are you an Arabic lady?
I just hope one day you'll do essence of real analysis.
Doing this course in my first-year undergrad, and I'm torn apart before even getting to derivatives.
Awesome videos...
I am falling in love with Maths until now I was just paying attention on solving problems but your channel videos are making me to see maths out of the box..
Where Maths can be applied in real world . Thanks for the videos.
Man thank you for all your videos, you make maths easy talking softly and explaining graphically.
this is like watching art. love it
You should really know better than to use \epsilon.
Of course, \varepsilon is what all _civilised_ people use!
Franz Luggin \varepsilon represent!
The thinness of the strokes in \varepsilon ended up fading a lot with the white letter on the black background, at least when the character was small, especially when the video was reduced to any lower resolution. You are right, though, there's something that seems a bit off with \epsilon.
I think the something off with \epsilon is that it looks too much like \in
The situation with the lunate (ϵ) and uncial (ε) epsilon is a travesty. Every teacher, article, and textbook uses them differently. Sometimes ϵ is used instead of ∈ ("element symbol") for set inclusion. Sometimes ε is even used for this purpose. Sometimes ε is used for small limits, or sometimes ϵ is used while leaving ε for indices. Honestly, nobody should use both forms of epsilon as different symbols in a single document; pick one and pretend the other doesn't exist.
EebstertheGreat as a greek we just usebone of these depending on what we prefer but using ε as the "element of" symbol is a true sin.
This is one his only videos where I’ve actually had a proper intuitive understanding... finally
Yes please on the probability series! When trying to learn it I encountered plenty of unintuitive (at first) concepts that are just begging for your clear method of explanation. A video on the normal distribution would be great to hear from you :-) thanks for your hard work!
7:47 That made me laugh.
Appreciate your sarcasm
I don't know why, but I cracked up at that part too
me too
Why are you stealing my account name and avatar?
You're a poser.
+Pears are Healthy
Hahah, me too (unironically)!
PI CREATURE PLUSHIE
Thank you for the intuitive description of L'Hopital's rule. As a university student I think that it was the most enlightening part of this video.
Same here. I'm pretty familiar with calculus, but when I was taught about L'Hôpital's rule many years ago, either the reason it works wasn't explained or I wasn't paying attention closely enough to take it in. So I've learned something, too. Thanks, 3b1b!
Pi creature plushies sound like a great idea, btw.
This also clearly explains why the rule only works in 0/0 indeterminate forms (and inf/inf, through some manipulations) despite the futile attempts of millions of calculus students through the ages who insist on "apply L'Hôpital first, ask questions later": Unless both f(x) and g(x) are zero at the point "a", then f(a+dx) and g(a+dx) are nowhere near equal to df and dg (sticking to the intuitive but informal use of d-something as a small nudge). This all works ONLY because the contribution from both f(a) and g(a) are exactly zero, leaving only the nudges.
BlueHawkPictures I study physics at a top uni in Sweden and my calc professors were the opposite. in my first intro course it said: no l'Hospitale. In the next course we got a full proof of the rule that involved lots of delta epsilon and Cauchy's mean value theorem.
The next thing our professor did after going through the proof was 3 examples of why we should basically never use the rule, including seemingly harmless functions that actually didn't give us the correct limit compares to other methods.
perhir01 yeah there are some cases where the rule just keeps repeating infinitely
I am very impressed with your videos. Looks like there is no limit in what you can teach us. Great job.
You are totally rocking it with these newer videos. It's not just nostalgia either. I love your earlier videos. They are amazing. Seeing actual geometry explained by calculus using animations is a true stroke of genius. I imagine these new calculus videos were inspired by your deep learning research.
I would love to buy your merch, but I haven't had a job for almost a year and a half. I have 150K (and growing) in student loans and just completely drained my 401k of 60k just to make payments and pay rent. I have no idea if posting this on UA-cam poses any risk. However, if you are looking for inspiration in your future videos. I would appreciate any financial mathematics animations you would dream up. I am not the same person who took out so much money. I however still a person. Thus, I need all financial advice I can consume.
I don't think there are haves and have nots, I think each person's finances are either accelerating up or down. The rich get richer because they are accelerating up, the poor get poorer because they are not accelerating up consistently. Right now, I am still accelerating down. Compounding interest is a lot like gravity.
One last thing. If you were to self publish a book using some of these animation frames. You might usurp all other math books because you would have a link to the videos the pictures are from. Thus, giving the reader the option to test their knowledge on the text and if that is not enough intuition then they would have the convenience of watching these artful videos.
6:22 that 'come on' is so relatable😂❤
Ismir Eghal by trep loop
Sorry, my three year old nephew got my phone
It is in 6:20
Add some more t-shirts with Pseudo-Hilbert curves, or the contour lines of the Riemann-Zeta function to your store.
I would absolutely buy this, 3b1b.
An invaluable resource for both tutoring freshman calculus and meditating on basic concepts in preparation for graduate level analysis.
I know calculus and analysis are similar to each other, but it'd be so great to see a series about real analysis like this as well! You are the best!
I don't know why, but I have learned about derivatives and limits a bit different in my country.
We started with limits to be able to define derivatives and we were solving them differently, which really showed the beauty of it. We mentioned l'Hopital's rule briefly, but I had to see for myself how useful it is in harder limits.
Also we didn't write the fractions df/dx at the end of everything, the calculations were much more clear thanks to that. However, we started using it at the end of integral, to know what is the variable. I guess we would have used in normally if we were planning to get to multivariable function derivatives.
Also, implicit differentiation was a bit different too.
This series is great, but it really shakes with my view on calculus, that I have built for the last two years.
Judging from your way of writing (in particular the commas before "but" and "that"), I would've guessed that you were German like me which is apparently not the case.
However, it is the "usual" way of introducing limits like you said since it is a quick way to transport rigorous definitions. Unfortunately, in doing so, much of the intuition is lost or hard to associate with the definitions.
(And one loses the reason why there is a dx in the end of the integral).
As soon as one gets to multidimensional calculus/analysis, it is quite important to give directions via df/dx...
They taught is the wrong way round in the UK, power rule and chain rule with some vague mention that limits is how it all hangs together but no detail.
If I keep pushing my upper and lower body to the limit, I might end up in L’Hospital... :D
Why isn't this pinned? I laughed so hard at it!
These videos are worth watching and very informative. Thanks for clarity in concepts
Ooooh I'm so close to the integral video. Thank you for the series!!!
Dropped out of Calculus my senior year in HS because I didn't know what was happening. Your videos make it so much more intuitive.
4:45 perhaps an analogy?
Imagine you have a friend whom you have no idea where she lives onand you want to meet her, though you're not allowed. Met at college.
So you ask a friend, and she points out it's in the next city over, so you head there.
Then arriving at the city, you ask another friend, she says it's in a particular neighborhood, so you go there
Finally you ask again and you're told on what street she's in.
Now you can't go there yourself and go into her house, since her father so grumpy, but at least you can see from a distance where the house is.
Limit is seeing the house in a distance and saying I know enough
And infinitely small is when you're welcomed to the house.
Both scenario leads to knowledge of the location but
You don't need to enter the house to know where it is, so you wouldn't need GPS to tell you next time.
*I know it sounds creepy but it's a weird analogy I come up with that might help?
Like limit is the direction of which everyone is pointing at
The GPS is manually checking actual values nearing up to the location
I just want to say one thing. Thank you so much. People like you makes me love maths again.
I don't have enough words to say how grateful i am for this video.....god bless you seriously
I'm using this to prepare for my upcoming bachelor degree in maths! I am done with my Abitur (somewhat like a HS degree) and start going to university in a few weeks
Hey! Hope you're enjoying your course :)
I'm a first year maths/science student too. I empathise with how it must be taking classes at home... I don't know how that's going on for you in Germany, but here in France most of our courses are remote.
Anyway, I'd very much like to hear what you're studying right now! We could compare ;)
i'll sag my teacher to show these videos whenever we will start studying calculus (4 years :P), books just give definitions over definitions, but this is very intuitive
Thanks a lot for making such great animations. I am truly grateful for your work. I probably would have spent an hour trying to understand the epsilon-delta definition if it was not for your video. (PS: I am a fresher in college) THANKS AGAIN
This series is so good. THANK YOU
Any plans to create an essense of stats / probability playlist?
Great series and animations. One thing:
Sure there is "something new on the conceptual level"! What you show is not a formal definition of the limit "L := lim{h->0} e(f, h)" in the way of "x := 2+3" (or "x:=SS0+SSS0" in Peano arithmetic) or "f(x) := x^2". You can't easily tell beforehand if a limit exists and thus provide a domain. The logical sentence "lim{h->0} e(f, h) = L ⇔ P(L, f)" is as close as you get to a "definition of the limit" and it implicitly captures a number or it doesn't. Everything else (e.g. writing, for general f, df/dx followed by an equal sign) is abuse of notation that you don't find in a logic text where all statements are closed (by universal quantifiers).
By the was h is often the "Hilfsvariable" (helper variable) in German text. The Planck constant was termed the same by Planck.
Keep up the great work.
i love you 3blue1brown, thank you for making and uploading these for free!
I just gotta pause the video halfway just to type this so I can tell how great your videos arrreeee. Bless youuu brotherrr
I really think this should be before derivates, it would make explaining them a little easier.
Fisher9001 I think in the derivative video he introduced a primitive notion of a limit which he called a "best constant approximation".
That was one of his intuitions for the derivative, not the limit - the value of the derivative gives the slope of a line that is tangent to a function at that point, and that line is the best constant approximation of the function at that point.
Who always read that as "hospital" like me or am I the only one??
Anyways, thanks for this brilliant video.
me to lol la hospital rule
The ô means that in old french, it word be os, so saying it "l'hospital" would be accurate
I think I’ll watch this whole series, your explanations are great!
I have spent almost 24 hours struggling to understand epsilon delta definition on whole of the internet but couldnt get feel of it. Thankyou so much u explained it in fraction of minutes .
I think I had never understood l'Hôpital's rule before... And I wonder why...
Anyway, thank you ! :D I love your videos ! :)
please include your logo and a math puzzle/oddity on the shirt. The logo looks like a mesmerizing eye, very visually interesting
it actually _is_ (represents) an eye (in fact 3blue1brown's eye is 3 parts blue and 1 part brown)
AMAZING series. I've never been so captivated
I LOVE YOUR VIDEOS SO MUCH THANK YOU!!! I was struggling to understand L'Hopital's rule and you cleared everything up for me. Thank you so much!!
Great video! I'm curious -- what are your thoughts on nonstandard analysis, which made "infinitesimals" rigorous? I thought your earlier videos in this series were very infinitesimal-friendly, so I'm kind of surprised that you're taking a stance against infinitesimals here.
Jonathan Castello Agreed; while I know limits were developed as a way to avoid working with infinitesimals, I nonetheless find the ideas of calculus more intuitive with them vs without. I mean, dx as an infinitesimal was the way calculus was originally done if I recall correctly (indeed Leibniz conceptualized dx as being an infinitesimal).
But yes, would looooove to see a 3b1b video on nonstandard analysis, or complex analysis!
Jonathan Castello I'd also like to here Grants perspective.
So far from the professors I've talked to they've suggested that non standard analysis is a different side of the same coin containing standard analysis.
it's an interesting idea nonetheless
I can see why he ranted a bit, though. A lot of people get hit with infinity for the first time in calculus, and it's too new and strange a world for them to keep up (as if the definition with a triply-nested ∀-∃-∀ formula is any less complex... but that's a different discussion). Any attempt to draw an infinitesimal means zooming in by some infinite amount, which just puts the "this diagram not to scale" disclaimer to absolute shame, and it just ends up being really unintuitive and un-visualizable.
Personally, I think infinitesimals are beautiful mathematics, but then again, I got introduced to infinite set theory when I was 11-12, so I've had a long time to develop intuitions for them. What I'd love to see is an "Essence of Infinity" series that covers Hilbert's Hotel, cardinal numbers, ordinal numbers, non-Archimedean fields, Cantor's diagonalization, maybe some strange consequences of the Axiom of Choice, and so on. But only if something like that were part of a school curriculum could I get behind teaching calculus with infinitesimals.
Okuno, I would totally be down for that as well. Seems like his next series is probablility (also looking forward to that one), but some time on infinites and set theory would be great.
Come to think of it, the nonstandard vs standard analysis conversation is alot like trying to pick whether you like playfair's postulate or Euclid's 5th better. If they're equivalent statements then the range of results should be similar.
That bernoulli dude got some wild, wild hair
Best explanation of l'Hopital's rule i have ever seen. Bravo, signore!
I've never seen any video that explains L'Hôpital's rule *intuitively!* This video is awesome!
Sometimes you just look at these videos and feel the need to give a clap...
7:46 lmao did that make anyone else laugh
Jup that was funny
This has helped provide some intuition into some concepts that weren't clear to me, much appreciated
I loved this presentation about epsilon delta definitions. Great work!