Limits, L'Hôpital's rule, and epsilon delta definitions | Chapter 7, Essence of calculus

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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[bluebrown]

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.

[klobiforpresident]

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.

[polychats]

I did not know there was this much good in the world.

[bmac]

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.

[txikitofandango]

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.

[GargaGaming]

Literally Everyone: You can't teach calculus to grade-schoolers...
3Blue1Brown: Watch me.

[sebster]

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.

[ruchirrawat]

How should the future generations pronounce your name ?


L'Hopital : yes

[ArpanD]

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.

[wamsang]

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"

[johnhippisley]

Thanks to your videos I achieved a 100% in my AP Calculus course!

[blzKrg]

Content like this makes me confident about the fact that the good part of the internet far outweighs the bad part.

[atharvas]

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!

[scottb]

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.

[Gamiboi]

I love how every once in a while, rewatching this series gives me different insights on calculus.

[crpr]

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.

[raunakdas]

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.

[PaulJamesOnGoogle]

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.

[wlxdncr]

Turn on the subtitle at 5:36. A little joke from our great teacher.

[jibeneyto]

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!!! :)