Limits | Chapter 7, Essence of calculus - Videos


Formal derivatives, the epsilon-delta definition, and why L’Hôpital’s rule works.
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  1. Next up is integrals. Follow the full playlist at

    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.

  2. 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!

  3. Will be great if you cover topics like laplace transform and fourier transform (I had problems understanding the last one but a really strogn tool in applications like signal processing) but this will be the essence of calculus, so I do not expect to see these Theme, anyway great work, this videos are awesome

  4. 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!

  5. all these visuals are so nice, what software do you use to visualize this, I would like to try something like that my self while learning math but redrawing all that by hand in different scales could get a bit tedious 😀 Did you program these animations?

  6. 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. Wonderful series. But I can't help by feeling that a lot of math education is complicated unnecessarily by the refusal of mathematicians to fully discard logicism.

    There is absolutely nothing wrong or illogical with accepting a notion like a function approaching a value as dx approaches zero as an empirical "truth" (i.e. accept it as being true). The problem is that feeling that there should be some logical basis for it. There can't be. There is no way to break that empirical "truth" down to a finitely formal truth in the strict logical sense. It's essentially the same issue with the 0.999…=1 "proofs", there is absolutely nothing with accepting this a true, but the notion that you can logically prove it within mathematics by moving numbers around algebraically is categorically illogical. The only thing that happens is a gish-gallop of overwhelming one's intuitive sense of what it means to be logical, and that's a bad thing. Things like these just a one level (or however many levels down) way of describing the same thing, a thing that must ultimately be accepted without recourse to logic.

    What Goedel and Tarski showed is that logicism is wrong, you might as well invoke sky faeries and magic teapots to prove things, logic is not answer you are looking for. Logic only kicks in after you have accepted a notion such as the limit, and if you think you are accepting the notion because of some underlying logic you are abusing the notion of logic to the point where it is no longer useful after you have accepted the notion of the limit. Save logic, reject logicism!

  8. If sinx/x as x approaches zero has has a limit:1, does it mean that numerator and denominator shares the factor (x-0)?
    It seems more legitimate when looking at the Taylor expansion of sinx.
    By the way can you ask a question related to the topic for us to ponder after learning it? So it will be more inspiring!

  9. I'd wish you would do a series on anything EXCEPT probabilities. Not only is probability the most boring subject, it is also the only one which doesn't help you understand any other subject.
    On the other hand there are so many interesting things you could do a series on, all the series which would be direct follow uos to this one (real analysis, differential equations, multivariable calculus, complex analysis, topology, differential geometry) or all the algebraic ones (group theory, ring/field theory and more linear algebra)

  10. I really don't understand the craze about l'Hôpital's (Bernoulli's…) rule. It's a fairly weak tool compared to limited developments, considering it's a limited development of order 1

  11. 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!!! 🙂

  12. I can't believe that so many people disliked this video! Even if you're a calculus master, you have to respect 3Blue1Brown's amazingly comprehensible explanations of indispensable topics.


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