Epsilon-Delta Definition of Functional Limits | Real Analysis

We introduce the epsilon delta definition of the limit of a function. We will explain the definition of a functional limit in depth, see some visualizations of it, discuss the negation of the definition of a limit, and then complete two epsilon-delta limit proofs for practice. #realanalysis

0:00 - Intro
0:59 - Epsilon Delta Definition of Limit of a Function
7:22 - Negation of the Definition (Function not Having a Particular Limit)
9:25 - Epsilon Delta Limit Proof 1
13:40 - Epsilon Delta Limit Proof 2
19:28 - Recap
20:05 - Epsilon Delta Limit Problem
20:54 - Outro

Oh My God Continuity is Weird: (coming soon)

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Thanks to Loke Tan, Matt Venia, Micheline, Doug Walker, Odd Hultberg, Marc, Roslyn Goddard, Shlome Ashkenazi, Barbora Sharrock, Mohamad Nossier, Rolf Waefler, Shadow Master, and James Mead for their generous support on Patreon!

Outro music is mine. You cannot find it anywhere, for now.

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[raptornomics]

Thank you so much! I'm self-studying Calculus after going through Algebra and Trig, and this came up relatively early on. Needless to say, I had a hard time understanding it at first as the only things I had ever "proved" up until that point were Trig identities. After watching your video (at least a couple times, I admit), I feel like I understand it much better. Also, happy to say I completed the practice problem on my own too before checking my proof against yours. (Look at me, Mom! I did it *all by myself*!) Thank you!

[guscastilloa]

Oh my gosh your explanation was amazing. I enjoyed every bit of the video, plus finally understood the proof that my professor did :) kudos for this amaazing series on real analysis, u've won a sub ;)

[desertpointshacks]

Hi there. I just want to say your videos are so fantastic and we are very lucky to have someone that can explain these topics so clearly and eloquently. Thank you so much for making them and please don't stop. I have watched this video a number of times - so good! But I have to ask - at 14.51 is the background noise from a fiend or a friend lol? Very curious to know when I get to that point in the video! Thank you again. I studied these topics years ago and it was always tricky because our professors were fantastic mathematicians but not always natural teachers (plus no you tube or rewind button in those days). You make it all so clear and understandable!!

[kingbeauregard]

I find it's super helpful to think of it like this. Suppose you're trying to prove the limit of f(c) = L. So, imagine a rectangle centered at the point (c, L) that is tall enough that the function never touches the top or bottom edges of the rectangle. Can you shrink the rectangle down to nothing -- no height, no width -- without the function ever touching the top or bottom edges at any size? If you can do that, it shows that, the closer you get to y = L, the closer you also get to x = c. And, that proves our limit.

All the math is just a matter of mathematically representing that rectangle, whose height is 2*epsilon (that's L plus or minus epsilon), and whose width is 2*delta (that's c plus or minus delta). So it's all about showing that you can establish a relationship between epsilon and delta, such that you get rectangles with the right geometry.

Turns out most functions are boringly continuous, unless there's an obvious divide-by-zero or if the function is defined with a discontinuity. But, someone's got to actually prove that functions are continuous, and epsilon-delta's the way to do it.

[punditgi]

Very thorough explanation. Well done, sir! 😊

[SkinnyMMA]

You reached 85k by the time I was watching this video.. So I guess some congratulations is in order!!! 🎉

[krasimirronkov]

Can you make a video explaining the definition of definite integral in depth pls

[instrumental]

At 2:47 shouldn't we say that f(x) itself converges to L but not the limit, since the limit is L itself?

[Blackpanter]

Do i need to show the scratch work on my exams, or is it enough to just write down the proof?
Cause then i dont show how i found what delta was gonna be, but it works...

[siriuss_]

I'm having trouble in showing that a limit doesn't exists, suppose that f(x) → 4, as x → 3, where f(x) = (2x - 1). How can we show that this limit doesn't exists?

[alondrachavez]

Hi, I am having trouble understanding how to place an upper bound on the Epsilon-Delta Proof 2. Can someone please help me? I have asked several friends and my professor and I still do not get it. I believe the confusion for me starts at 15:00 of this video.

[kranthisingh]

Dude. Your content...Fuck. THANK YOU !!

[minamishi]

So this is my informal paraphrase of the delta epsilon definition: "if the limit L exists at x = A, no matter how small epsilon is, there will exist a set of x values, which satisfies the condition |x = A| ≤ delta (where delta is a value we have to find but we know it exists) so that |f(x) - L| ≤ epsilon"....is that correct or close to correct?

[BedashrutiRay]

What is the reason to exclude c(limit point) when we write the defination

[lifeforever]

Is approaches or closer and closer the right term ?