Real Analysis | Precise definition of a limit.

We introduce the precise definition of a limit, given an outline for an epsilon-delta proof, and show some examples.
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[@JaybeePenaflor]

Ah, the wonderful memories of when I first encountered the formal definition of the limit have returned after watching this video!

[@moncefkarimaitbelkacem1918]

Wonderful???? It broke my dreams, don’t show this to kids

[@JaybeePenaflor]

@@moncefkarimaitbelkacem1918 LOL 😂

[@ibrahimbelkadi8715]

@@moncefkarimaitbelkacem1918 hhhhhhhhhh fr fr

[@elgourmetdotcom]

It would be great if you could do some more challenging examples using the definition such as for sin x / x or complicated functions since no examples of those are usually available in books, always the same basic ones.

[@elgourmetdotcom]

Thanks guys but I’m not saying I need to solve that limit, I’m merely suggesting it would be interesting (given the type of problems Michael solves) to include some more challenging examples. That’s all.

[@zhangkevin6748]

@Λ when you learn about epsilon delta proof you can’t use limit laws, because they are not true to you. They are only true after you epsilon delta prove them. As for squeeze theorem the same logic applies, you need to be aware that the question specifies using epsilon delta definition.

[@FredHandle1]

What I'm wondering about is that there should be an archive where the math society keeps record of all difficult proofs by subject. Does one exist? if you know pls advise. And yes if you are a student of math, one needs to work his hand & mind at proofs to get the feel & be creative. But in related fields such as fluid flow, pump design etc, it is often the case that an industrialist at short notice needs a ready solution without having to rediscover it.

[@alishams6479]

This video shows a very clear explanation of the limit of the real functions for beginners who have hard time to grasp how this exact definition can be linked to the intuitive concept of the limits. However, it is very important to exclude x = a in the neighborhood of a. Therefore, the condition on x should be written as 0 < |x-a| < delta, which conveys the idea that the value of the function at x = a does not matter or x = a may not belong to the domain of the function.

[@muwongeevanspaul9166]

I just love the humility you posses and the beautiful explanation u possess. U are a natural teacher. Thanks for your lovely lessons.

[@Luffy_wastaken]

I literally understood the definition days ago and yet I am seeing as many videos for this as possible- I just don't get satisfied with my understanding 💀

[@dukeofminecraft]

Finally a video on how to precisely write epsilon delta proofs. Thanks for this, most the videos aren’t nearly this structured and everyone has a different method.

[@The1RandomFool]

Your shirt is inside-out!

[@joshmckinney6034]

Omg not you focusing on the wrong thing!!!

[@CarlosFloresP]

@@joshmckinney6034💤

[@duskRenn]

you're a REAL lifesaver, man. thank you so, so much for the clear and detailed steps

[@hokou8766]

A very clear explanation, easy to understand for a beginner. Thanks for your sharing. It's a very nice video, especially for a freshman in maths.

[@yoav613]

in the last example you can also take delta to be equal to -4+sqrt(16+e) and you get(x-4)(x+4)isless than 16+e-16=e

[@VeteranVandal]

I think follow this with the definition for n dimensions is very helpful to make people understand that the delta(epsilon) might be hard to find, but if there's one, it's enough, otherwise the counter exemple tends to be easier.

[@mystified_dignity]

I love the lucidity of Professor Penn's videos.

[@pup4301]

Just learned this in Calculus class.

[@mikehughes6582]

You explain it much better than the professor I had. Luckily, there was a smart guy in class that would explain it.

[@DarkMonolth]

My calc 1 professor introduced the formal definition with the idea of target shooting. Not sure why that’s stuck with me over 4 years later, but maybe it was repeating the definition over and over in my head about 700 times

[@YorangeJuice]

Thank you so much, explained better than anyone else I’ve watched cover this topic

[@leodijkstra6666]

I can’t watch these kind of proofs without constantly hearing Tom Lehrer singing that there is a delta for every episilon :D

[@DumblyDorr]

This was 1st semester math, 3rd week real analysis I ... I'm like 99% sure around 99% of students at that point are *not* gonna type up their homework in LaTeX. :D ... I mean, I do - or I try to find the time. But this ain't my first rodeo. Even if you can write LaTeX well, writing well-formatted proofs with correctly justified multi-line steps - if possible in some proof calculus, or if you need to include some diagrams... that's really a lot more time-consuming than just writing it down by hand (/with pen on a tablet). Still - pays to learn to do it early.

[@juliapezant9768]

Such clarity. Thank you so much for doing this for us.

[@newbie6036]

on 11:33, why do we use the |x+4| 7? If we want to replace the |x+4| in inequality |x-4|*|x+4| < \epsilon, shouldn't it be 7 * |x-4| < |x+4||x-4| < \epsilon. Therefore |x-4| < \epsilon/7?

[@gametimewitharyan6665]

Wow this was such a good explanation, thanks a lot, I was struggling a bit to fully understand the epsilon delta definition

[@punditgi]

Excellent starter examples!

[@isaacfernandez2243]

Something it's very pleasant about him writing with chalk.

[@lailoelmuratova67]

this title used to be hard to understand but now I well understand it thank you for your interesting lesson

[@azulpurpura1230]

Excelente video as always, if you can (and want) make more proof for a little bit complicated functions, that would be nice :3. Greetings from Mexico :), I can't afford a math collage so this is really nice u3u. Thanks again :)

[@TateVanPatten]

wonderful that you are learning for the joy of learning!

[@chancerowland5585]

Hey Mr.Penn your shirt is inside out ...

[@roberttelarket4934]

Well what do expect from an outstanding absent-minded mathematician? He cares about the subject matter not worldly things. You are very observant and for that reason you're probably a mathematician.

[@rcbuildmaster6192]

Good explaination. Thank You very much Sir.

[@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?

[@lueelee6063]

i needed this

[@roberttelarket4934]

The delta-epsilon definition for the limit is the most important concept and event in the history of the world bringing precision and rigor; brought forth by Bolzano, Cauchy and its final form by Weierstrass.

[@mendelovitch]

What further developments did it enable?

[@roberttelarket4934]

@@mendelovitchPrior to the delta-epsilon definition the limit was used by faith not rigor that not only math was on shaky grounds but everything from physics and chemistry that relied on it was as well and the real results from the latter two that created the modern world. This is not to say that there would be guarantee that their theoretical creations implied that in fact is how the physical world works. Math just helps them produce various models and they choose if possible the one that best fits the observations. The specific developments you asked for are more than thousands that you can find and see in their discoveries for almost two centuries!

[@karolakkolo123]

@@roberttelarket4934 yes but saying that's the most significant thing out of all formalizations is quite the claim

[@rogierbrussee3460]

Here is an alternative definition of limit that is also rigorous. Call a function $k:[0,\infty] \to [0,\infty]$ a control function if it is non decreasing (ie. if $ \delta_1 \le \delta_2$ , then $k(\delta_1) \le k(\delta_2)$, and $\inf_{\delta > 0} k(\delta) = 0$ (i.e. if $e \le k(\delta)$ for all $\delta > 0$ then $ e\le 0$.). We say that $\lim_{x \to a} f(x) = L$ if for all $x e a$ in the domain of $f$, we have $$ |f(x) - L| \le k(|x -a|) $$ for some control function $k$. Note that control functions may be infinite so this really is only a condition for $|x -a| 1/2$) is a control function showing $\lim_{x \to 0} \sin(x)/x = 1$. To make contact with the standard definition of limit in the video: using the above one sees that for all $\epsilon > 0$, and for $\delta = \sqrt{\epsilon/\sqrt{2}}$, we have for all $x$ with $0

[@ZanderzMcCluer]

Where does the |x+4|

[@zatee6553]

x is at most 5 thus |x+4| is at most 9 (we are only interested in the upper bound)

[@ellenancowitz1808]

@@zatee6553 thanks!

[@NattyPi]

@@zatee6553 wow thanks for that explaination

[@goodplacetostop2973]

14:20

[@ntvonline9480]

Thank you!

[@glennxhose7217]

you make seem so easy🤛🤝

[@JonCurryIsolatedGuitarist]

I'm so impressed by this sir. The amount of vids I've watched on this are too numerous to recall. This is by far the absolute best presentation I've ever seen on this subject You know your shit, that's for sure. Well done man!!!!

[@padraighill4558]

the definition is missing two things. a must be a limit point of the domain (of course R contains all of its limit points but this is not always true). also, it is a DELETED delta neighbourhood about a. you say this in words but this is not in your definition.

[@omriman123]

Am I the only one noticed that he wore his T shirt reversed after 8 min into the video?

[@mendelovitch]

שלום!

[@omriman123]

@@mendelovitch Hello there🤣

[@anshumanagrawal346]

Haha

[@reemahmad8381]

thanx

[@doontz111]

I loved RA 1. It was such a great class

[@mercykyalo2357]

Nice one

[@beatoriche7301]

Minor question: Is it common in the US to practice ε-δ proofs using concrete examples? The analysis textbooks in my mother tongue start by directly proving limit properties from the definition, which makes these types of examples pretty trivial - for instance, for the second one, the claimed limit follows directly from the continuity of the identity function and the multiplicativity of limits. Obviously, both approaches are valid, and I'm not really sure which one I prefer - I just wanted to note this because I found it interesting and ask for the perspectives of other people. From a didactic standpoint, practicing with these types of specific examples might help acquaint students with estimations - which are obviously a very fundamental tool in analysis -, especially if the ε-δ definition of a limit is introduced early on in the course; however, analysis courses already feature a lot of those estimations early on, so it might be a good idea to move directly to proving theorems rather than getting hung up on calculations.

[@xriccardo1831]

Both approaches are important.

[@stephenbeck7222]

It’s good practice to be able to use the original limit definition. Of course we quickly move on to bigger things.

[@beatoriche7301]

@VeryEvilPettingZoo Here's the thing, though: you'll see a lot of ε-δ proofs in an analysis course either way. Going through the proofs of limit properties, not to mention proofs of theorems about continuous functions and other things you'll encounter later on in an analysis course, should yield enough structural understanding of ε-δ proofs to write one yourself. Not to mention that a lot of textbooks will leave the occasional ε-δ proof as an exercise, which means you'll get even more practice - in addition to ε-δ proofs you'll encounter in other courses. And if the analysis course introduces limits of sequences before ε-δ limits (which, in my experience, most of them do), there's really not much to learn because they're essentially the same on a structural level. It might be a different story with calculus courses (in my country, we don't have anything analogous to those - math majors generally take analysis as one of their first courses), but in analysis courses, I don't know if this type of practice is really useful.

[@JesusHernandez-xv7lf]

For epsilon < 0

[@CharIie83]

I have a need to be handheld too much for this channel I think, yet anyway

[@mendelovitch]

I would mention that in my next date if I were you. Long moonlit walks on the beach, Italian food, and holding hands while watching Michael Penn videos.

[@stefanieblumenthal3105]

Is it okay to introduce a domain where the function is not defined? If I have f(x)=1, for x an integer and f(x) undefined for other x. Eg, 1,1,1... Can I just still make delta as small as I like, say 1/2, in the undefined domain?

[@samirt1976]

very well explained…. 💯

[@beginneratstuff]

11:30 I don't understand the logic behind replacing |x+4| with 9, because it was found that |x+4| < 9 not that |x+4| = 9. I thought about it for a while but I can't seem to figure out why it makes sense

[@ali_aldur]

This is one of the properties of the inequalities. you just have to divide the inequality by |x+4|

[@beginneratstuff]

@@ali_aldur Thank you. I eventually figured it out. I didn't have enough experience working with inequalities to know why before but it makes sense now

[@anshumanagrawal346]

The definition is slightly wrong, you need 0

[@kimjiimnida9212]

how did it happen that absolute value of x+4 is less than 9? I am confused.

[@rosasoiwerpuffgirls5806]

I am same

[@emaesee4284]

Yeah idk, -1

[@FrostDirt]

@@emaesee4284 since 3 < x < 5, the maximum value of x is 5. Therefore, |x + 4| < 9.

[@dnuma5852]

well think about it like this: suppose we have δ=1 and |x-4|

[@xaxuser5033]

but here in your definition u forced f to be continous at a which is not always the case

[@steves3948]

Cool.

[@ad2181]

The limit definition needs to include a "for all" x qualifier.

[@fullfungo]

No

[@AlessandroZir]

🔥🔥🔥🔥

[@roberttelarket4934]

It's very important in this definition with the wording. You must say for each "small" positive epsilon there is a CORRESPONDING delta... NOT for every epsilon there is a delta...

[@calibratingform]

The definition of a limit only requires epsilon to be positive, not that it be "small." It begins: "For every epsilon > 0, there exists delta > 0 such that ....." But yes, it is equivalent to the apparently weaker statement that begins "For every 'small' epsilon > 0, there exists delta > 0 such that ...." essentially because if the statement holds for a given epsilon_1 > 0, then it also holds for any larger epsilon_2 > epsilon_1. But to make this latter phrasing completely rigorous, one needs to clarify the meaning of "small," which would require introducing yet another (Greek?) variable. This technical annoyance is probably why virtually every textbook uses the former phrasing (instead of the latter).

[@roberttelarket4934]

@@calibratingform: True small is meaningless. Again the important point is that Mike and others use incorrectly the word for ALL epsilon there is a delta...This is vague and misleading. That is why you need to say for EACH epsilon chosen there is a CORRESPONDING delta....

[@calibratingform]

@@roberttelarket4934 You're objecting to the use of the word "all" as opposed to "each"? Mathematically, "all" and "each" are synonyms. Is your concern that "for all epsilon, there exists a delta...." sounds too much like the very different (incorrect) statement "there exists a delta for all epsilon...."? I mean, maybe, but it's the word order that really matters here. I suppose adding the word "corresponding" is psychologically helpful, though, sure.

[@roberttelarket4934]

@@calibratingform: Yes I am objecting! In 1968 as an undergraduate and junior taking the advanced calculus on an exam we had to use the delta-epsilon method. I used the word "all" in my response and was penalized 15 points from that question with a comment from my professor! I at that time 19 years of age luckily understood the concept that young but admitted I made a serious mistake. Since then I've known better! We shouldn't go by the English or any other Ianguage imprecise use of all and each. Sorry I don't agree with you, they are different in math. If we say for all epsilon there is a delta that could be construed by a beginning student that for all epsilon there is only a = one delta which is not correct!

[@calibratingform]

​@@roberttelarket4934 I see :-) Well, in today's 2021 mathematics, the words "all" and "each" are universally accepted as mathematical synonyms. The distinction is in the word order: "For each epsilon there exists a delta..." is different from "There exists a delta for each epsilon..." It is the second phrasing that you're referencing when you describe the misconception that there exists a common delta that applies to each epsilon simultaneously. By the way, the articles "the" and "a" have accepted meanings, too: "the" means "only one," whereas "a" means "at least one." I agree that this is all very confusing for beginners, but this terminology is now completely standard among professional mathematicians.

[@martinhazard2025]

No

[@thiagomilanezi1026]

👏👏👏👏👏

[@rialtho_the_magnificent]

Midroll ads. Why?

[@khalilmohammed2297]

but we know that a numbers in the neighborhood of a number is endless .

[@JB-ym4up]

If we take the limit of a function like f(x)=1 then delta will be 0 for all values of x and epsilon.

[@JB-ym4up]

@VeryEvilPettingZoo ok it clicked. For the constant function delta can be arbitrarily small regardless of epsilon, but not actually be 0. Thanks 👍

[@arthurleywin1366]

Where did that 9 come from?

[@arthurleywin1366]

Someone please answer because I'm pretty stupid but trying to learn because of my course

[@anshumanagrawal346]

x is between 3 and 5, so x+4 is between 7 and 9, but that means it's absolute value is also between 7 and 9, which means it is less than 9

[@robertobarrosofernandez5019]

First comment :)

[@jimallysonnevado3973]

I think this is too late to be placed in this series

[@stephenbeck7222]

He already did a bunch of limit/convergent proofs with series which is more important as a foundation for analysis in general. Now he’s starting to prove the big theorems of calculus which rely on the existence of convergent sequences in the reals to be rigorous, and so he picks up calculus with function limits.

[@nuclo7586]

14:20