New calc video! Limits.

I agree. Especially as Leibniz was thinking of infinitesimals when he invented his dy/dx and \int dx notations.

Great point! And in fact, an earlier draft even included sin(1/x) as the counterexample. I think I'll leave a small note on the screen at some point, maybe towards the end, for those inclined to pause, read and ponder.

3blue1brown

Very nice! Just a thought: this could give someone the idea that the only reason limits might not exist is a sudden jump in the function's graph. The video is pretty long already, but how about mentioning a function like sin(1/x) and saying that interested viewers can prove for themselves that it cannot have a limit as x->0 even though there is no "missing" range of values?

If I can nitpick, infinitesimals have a couple different self-consistent formulations, so I wouldn't call them "paradoxical", but even as a fan of them, I'd be totally fine with "confusing".

Okuno Zankoku

You handled the epsilon-delta part well. I was expecting you to go into it with more detail; but I was pleasantly surprised at how you just opened up the area, allowing the interested reader to pursue further if desired. And I love that little historical piece behind L'Hopital's rule.

Magnasium

Self-correction: just saw that you do move the horizontal line to both values. So scratch that point. Talking about approaching from one side may be worth a thought nonetheless.

Lionel Pöffel

Excellent. I've got one point: in the counter-example around 9:35 you display the horizontal line with the epsilon neighborhood *in between* the two values that the function approaches from left and right hand side, respectively. I think it may feel unintuitive to some. Maybe it's better to draw the horizontal line through *each* of the two values and show that finding a delta for an arbitrarily small epsilon neighborhood works for *neither* of the two values. You may also want to spend a few words on approaching from one side.

Lionel Pöffel

Thanks, I'll take a closer listen.

3blue1brown

Very nice! But, is there some variability in the audio levels? It seemed to drop a bit a few minutes in, and then maybe it came back.

Peter Su

Awesome! Something I don't completely understand is why the derivatives are not defined in "angles" in curves. I Just let here the proposal...

I actually went and asked her. I'm not in a great state right now...

great video!

The "just ask Siri" might date the video a bit eventually. And some people might not know what Siri is; it threw me off a bit to be honest.

Jake Palmer

Good suggestion, it always helps to be a bit more concrete.

3blue1brown

Oh interesting, I just went with the default LaTeX \epsilon. I guess I could swap it for \varepsilon everywhere. Thanks!

3blue1brown

Great as always! Insignificant nitpick: at 15:20: the upper arrow overshoots the graph a bit

Job van der Zwan

You did a really good job with those epsilons and deltas in the end! I think your way of handling it was the right call; the video doesn't feel overly long. I'm sure that anyone dealing with epsilons and deltas would try to look through that segment a few times. It might help if, in your visualization around 8:50, you wrote in "epsilon = 2", "delta = ____" and "epsilon = 1", "delta = _____". I remember that when I was learning this, I needed the epiphany of "oh, epsilon is really just any old positive number".

Benjamin Grossmann

This is one of my favorites so far! My first year at uni came rushing back, only this time much clearer ;)

fantastic video as always, one small detail: As someone who has taken ancient Greek lessons, this sign for epsilon, ϵ (or whichever you used in the video), hurts my eyes. This sign here, ε, looks much nicer.

Andreas Blatter

Really polished video, best of the series so far.