The hardest concept in Calculus?

[@Triszious]

Definitely a video that deserves more love. The ε-δ definition of limits is a very powerful tool in nonlinear dynamics and stability theory, so there's no doubt that it has important real world applications.

[@kroyboy]

Love this video! Showing how a δ-ball in the domain transforms to a "blob" in the image was a really nice touch. I do wish you could've visually shown what a function that fails the ϵ - δ test looks like, i.e. a function where the limit does not exist. But irrespective, great video. Been a fan since the umbral calculus video. Keep up the good work!

[@aurorazuoris6654]

a function where the limit does not exist would look like: even if the δ-ball gets infinitesimally small, the image blob converges to a certain size. Because of this, you can never get a blob that is smaller than the ϵ-ball for some ϵ by making δ-ball smaller, as the image blob just never shrinks below a certain size. This is why the proof states that it is *for all ϵ*. As the limit only exists if you can make the image blob as small as you want to. And doesn't if it only gets small to a certain extent and never smaller.

[@SafetySkull]

For some reason in my uni the first unit I encountered epsilon-delta proofs in was a multivariable calculus class. So, before doing limits of sequences or series - before doing limits of functions as they approach infinity - before even doing limits of continuous univariate functions as they approach a point, they had us doing epsilon-delta proofs in two dimensions and I had no idea what was going on. I passed the class somehow but that basically stayed as a knot in my brain for a year until we did epsilon-N_epsilon proofs for sequences in number theory. It finally made sense.

[@DynestiGTI]

Wow I wish I had this video when first learning Real Analysis! Love your voice, and the animations and music are really nice!

[@pedrovpa1]

Imagine having Morgan Freeman as a Calculus Professor

[@zaydmohammed6805]

Morgan?

[@pedrovpa1]

@@zaydmohammed6805 Thanks

[@1224chrisng]

wait did you write Gordon Freeman?

[@TranquilSeaOfMath]

Great video, explanations, and illustrations!

[@Bruno-el1jl]

Great video! Thank you for the thorough work!

[@SuperDeadparrot]

I think he’s talking about uniform convergence without actually using the words.

[@chennebicken372]

Can the distance Function be any metric?

[@mathematimpa]

The definition works for any metric. However, if the two metrics don't agree on which are the open sets (which is common if we use more exotic spaces instead of R^n) then they may give conflicting notions of limit. On the other hand, if they agree on which are the open sets, the definitions using each metric are guaranteed to be equivalent.

[@Walczyk]

calculus of residues and fractional calculus are both harder concepts imho

[@michaelzumpano7318]

This was dense!!! Very nicely done!

[@AviPars]

Great video and voice over

[@andersbahrami1649]

Did you play/write your own background music?

[@mathematimpa]

It's me playing. I improvised some arppegios and chords until I got something that sounded nice, then I edited it down to some loops.

[@shreyanmullick4102]

This video solved half of my life problems.. thankyou so much ❤️❤️❤️❤️

[@78Mathius]

I could apply the definition before. I really get the point now.

[@johnchessant3012]

Great video!

[@thatkindcoder7510]

It just hit me that showing that the topological equivalence of metrics is defined in an analagous way. For every d1-ball of radius r1, there exists a d2-ball of radius r2, such that the latter ball is a subset of the former. I've not yet reached the point where limits are defined on topological spaces, but would metric (topological) equivalence mean something there?

[@mathematimpa]

Metric equivalence would mean that both spaces have the same notion of limit. The way you'd extend this definition to a topological space is as follows: Let f:U→V be a function between 2 topological spaces. We say 'b is a limit of f(x) when x tends to a' iff for every open neighbourhood N of b (this plays the role of the ε-ball to the right in the video) there exists an open punctured neighbourhood M of a (this plays the role of the δ-ball to the left in the video) such that f(M) ⊆ N. Since metric equivalence means the topological spaces have the same open sets, its easy to see why they'd have the same notion of limit. Note however that we say 'b is A limit'. This is because unless the spaces have the Hausdorff property there is the possibility that the limit is not unique. Every metrizable space (where you can define a metric that would give you exactly the open sets of the topology) is Hausdorff, but there are some exotic examples showing the converse is not true, so you have some situations where you can talk about limits, they are unique but at the same time metrics make no sense.

[@pablolecce6931]

I suggest you to watch the video that Grant Sanderson did for the first SoME and you will see there some interesting things that could help you to do better videos. You still need to work in how to build your explanations, for example, the explanation of the circle being mapping to a point was difficult to understand ( maybe, for some people that knows about) but to me was impossible. Despite that, you are in the right path

[@gregoryk_lite]

I agree, the transition to circles was too fast, I had to pause and think about what he means by those circles shrinking, fitting and so on. Nothing bad in having to think, it's math after all, but probably someone who does not know the epsilon-delta stuff or does not have intuition about one object being mapped to another will have no clue what's going on. But anyway, this kind of videos is only good (unless you're watching it just as entertainment, or to get interested in an area you know nothing about) as an addition to a formal course, where you memorize very strictly all the definitions, solve tens of problems, and with all of that you watch the video and get a better intuition/a new perspective.

[@anshumanagrawal346]

5:26 this is incorrect, this is only true if c = g(b)

[@anshumanagrawal346]

Nevermind he talks about this later in the video

[@knutthompson7879]

epsilon-delta made perfect sense to me. Not hard at all.

[@blacklistnr1]

@0:56 "Must be the same no matter how it approaches" That is simply not true and it sticks out on this beautifully animated video. If my system completely breaks down on the other side of the limit, what's the point of enforcing this condition when I can freely use limit-related tools for the side I'm trying to describe/analyze? Even in your example the limit can be defined there as the set {red, blue} This is not a general property of the "core notion of limit" but rather of this very specific definition, not to even mention how the underlying assumptions about metrics are brushed under the carpet. There are many spaces, many uses and many definitions for limits. Again, great video, explanations and animations except for the "must be the same no matter how" without saying it's specific to this definition.

[@mathematimpa]

Sure, but we have to start somewhere. And that somewhere is real spaces with their usual topology. If I were to point what I think is the most fundamental idea behind limits that would the limit of a sequence and I think you don't even need to tie the definition to metrics, but can get away with working with just open neighbourhoods in Hausdorff spaces. For example, the limit of a function f(x) being L when x->a (in the sense of the video) is just the statement that for every sequence (x_n) that converges to a, the sequence (f(x_n)) converges to L. You can get lateral limits from this by just restricting the function domain, you can get different metrics/topologies by adjusting your notion of sequence convergence, etc. Anyway, this was aimed at people that maybe saw calculus but are not mathematical literate enough to understand what the ε-δ definition is really saying. So going down that rabbit hole would have been overkill.

[@blacklistnr1]

@@mathematimpa "not [mathematically] literate enough" That was me with a lot of parts of math for a long time, still am with many. I would have loved to hear about math's freedom of "we have these conditions but only because we choose them" a lot earlier. It's not a rabbit hole to convey that idea, just a small good intro of "There's many ways, but this one is quite interesting, let's see what it can do" and "make/choose because" instead of "must".

[@martinepstein9826]

Technically, the definition in the video allows for one-sided limits. e.g. suppose you're taking the limit of f(x) as x approaches 2 from above. This is the limit as x approaches 2 in the space (2,oo).

[@JackTheAwesomeKnot]

He's talking about a double sided limit. You are talking about a single sided limit. The terminology is important.

[@menjolno]

my wifi bad so i think i have to make this joke before it goes down. Yo mama is such a wifi that she auto reconnect with yo papa.