epsilon-delta definition ultimate introduction

[@blackpenredpen]

Check out 24 more rigorous proofs: ultimate calculus: 24 rigorous limit proofs kzbin.info/www/bejne/d5fVn4yJarhrjKc

[@redblasphemy9204]

back when my fiancé and I were at our 1st year of university studying analysis, we were hanging out together and it was a classic romantic dusk moment... and then I screamed "I JUST REALIZED HOW EPSILON DELTA WORKS!" he is still mad at me lol

[@blackpenredpen]

😆

[@verypotato6699]

how is he still mad? if i were the reason anyone realised how epsilon delta works i’d be overjoyed

[@fareschettouh]

Hello Please can you resolve this équation not geometricly (2^×)+(3^×)=(5^×) I know that x=1 but how you can find this solution thank you

[@iitguwahaticseairunder500r2]

@@fareschettouh heyy, Use a function f(x)= 5^x -2^x - 3^x Then draw the curve using curve tracing The number of times it crosses x axis is the number of solutions of the original question. I don't know any other geometric soln.

[@michellauzon4640]

@@blackpenredpen It is quite easy to figure the contradiction which is There exist epsilon > 0 such that for all delta > 0, there exist x such that ABS(x-a) < delta and ABS(f(x)-L) > epsilon. If so, we can put delta(n) = 1 / n for all positive integer, and pick x(n) for each n. Then x(n) tends to a, and f(x(n)) doesn't tend toward L.

[@BoazNahumPlus]

I learned 3 things today: 1. The definition isn't that bad, calming first does help. 2. That Pockeball has no use! the mic is just next to it. I'm shocked. 3. I noticed your huge stock of Expo's in the back for the first time. Thank you for your amazing videos and work, offering knowledge for free. Math can be hard as it is, and you help it seem reasonable.

[@blackpenredpen]

Thank you : ))

[@christianchavez2202]

I'm shocked too :0

[@lakshya4876]

''pokeball has no use'' MY LIFE IS A LIE

[@mikee-fl8ex]

it may act as a noise damper

[@JohnSmith-rf1tx]

This is great. Could you also do a counterexample where the limit doesn't exist and show how it breaks using the epsilon-delta definition? I often find that showing a counterexample that highlights what goes wrong is often more helpful in building understanding than just seeing one more example where everything goes right.

[@henriqueassme6744]

That's what I was going to ask. I have no idea what happens when the limit doesn't exist using epsilon-delta definition

[@kingbeauregard]

y = sin(1/x)

[@kingbeauregard]

@@henriqueassme6744 I believe what happens is, you reach impasse with the arithmetic: you find yourself at a point where there is no way to get to an expression like "delta*constant = epsilon".

[@popularmisconception1]

It means that for some too small epsilon (desired output error margin), you just cannot find small enough input margin delta that guarantees you to fit into that desired output error margin epsilon. You can't, anyhow close on the input side, the output will always be too off. Think about a threshold function for example: Zero or grater -> returns one. less than zero -> returns zero. What is the limit in zero? There is none, because whatever anyone would claim it to be, the values of f(x) around the zero will still be zero on the minus side and one on the plus side. Even if you chose it to be one half, the minimum output error you can get is one half even for infinitesimally small difference from zero on the input side. Even if someone told you it is one (i.e. the value of the function by definition), if you approach that value from the left, you are still too off from the alleged limit.

[@lostwizard]

This is exactly where my calculus professors in university went wrong with the epsilon-delta explanation. They concentrated almost entirely on cases where it works and failed do more than a cursory "and we see how it doesn't work in this case" after a flurry of barely legible scribbling for a couple of counter examples.

[@coreymonsta7505]

For understanding the definition, it helped me to think about the absolute value parts as distances. Ie read |x - a| as "the distance between x and a". This even makes the more general definitions pretty digestible, because distance is what it's all about.

[@sadiakhan6500]

saying that "this is one of the hardest things in Calc 1 and a very difficult thing to explain" honestly made me feel so much better, and I actually gained an understanding through this video. All the videos that i've watched that try to build an understanding didnt help, even if they used visuals, but this video's explanation in a more algebraic format helped SO much, and just admitting that it's not easy just makes it feel more like I'm not alone in struggling to understand the logic behind this proof. Like I knew how to write it, but not what it meant. Now I know both. Thank you so much!

[@aiosuu3507]

the king of math has uploaded :0

[@agrajyadav2951]

Euler or Archimedes?

[@ojaskumar521]

@@agrajyadav2951 Bprpmedeseuler

[@outofnowhereboy8448]

I have a masters in statistics and a degree in maths and at uni this was the only module (not exactly called calculus but the module that contained this element) I failed, retook and STILL failed. And I put it down to, my stats teaching was AMAZING (hence why I followed stats) and the “pure maths” teachers just did not care to try and show any kind of examples to explain things. My point is, all this time later, and I have finally seen some teaching where it goes outside of “here’s this definition, if you don’t understand, you be stupid” and has bothered to put some actual real world understanding to it, that I finally get it. This is an amazing video and I respect it soo much. What a great example of how maths should be taught!

[@blackpenredpen]

😊 and thank you.

[@AdoptedPoo]

i hate stats, but my stats teaching is garbage. sucked the life out of learning because it was so boring.

[@marcelandrade5759]

I'm a Brazilian engineering student and I'm learning calculus with a professor from another country who speaks better English than my professor at the university, who is also Brazilian and speaks my native language. This is amazing, this video helped me a lot. Thank you so much

[@matheusreidopedaco]

This is the only explanation of the delta-epsilon definition of the limit that I could understand and now I got addicted and can't stop proving limits! Thank you so, so much for this video.

[@michaelsekeleti4652]

Great I can be glade that u can help me

[@blackpenredpen]

εN definition (finite limit at infinity): kzbin.info/www/bejne/b3uwd3-wfdmfoKM

[@kingbeauregard]

I've spent a lot of time chugging through epsilon-delta this past month, and I think I figured it out AND the explanation that would work for me. The trick is to stay away from the numbers until the concept is firmly in place. SO: imagine that you're trying to prove the limit of a given function at (a, L). Can you draw a rectangle around (a, L) that is tall enough that the function never touches the top or bottom edges? And, can you scale that rectangle all the way down to nothing such that the function never touches the top or bottom edges? If you can do that -- if you can derive dimensions for the rectangle that make it possible -- then since the rectangle scales down to converge on (a, L), the function must too, and that proves the limit. Our rectangle has a height of 2*epsilon and a width of 2*delta. So the math is all about proving that you can write epsilon in terms of delta, and probably as a straight linear function. You will start with two inequalities: |x - a| < delta, and | f(x) - L | < epsilon. Then you get to work on the latter. From there is it mostly a matter of basic math operations involving inequalities, but with one additional thing you can do: you can replace any term on the left with a simpler expression that always makes the left side larger, or at least never gets any smaller. So we are treating epsilon as an elastic term that we can make as large as we need to, to compensate for whatever shenanigans we're doing on the left. It is also usually necessary to restrict our x values to a narrow region around "a", which is fine, because we're primarily interested in what happens close to the point (a, L). Very often the "simpler expression" and "restrict x" steps happen together: "I can swap in suchandsuch simpler expression, but with the understanding that x will stay within a narrow region that makes it mathematically valid." Now, remember that the goal is to write epsilon in terms of delta, and we've already said that |x - a| < delta; so, when you've got things to the point where it's |x - a|*(some constant or other simple expression) < epsilon, you can swap in delta, and it becomes delta*(some constant or other simple expression) = epsilon. Once you do that, you've got your simple relationship between delta and epsilon, and you've won. You have proven that it's possible to draw a rectangle around (a, L) with dimensions such that, when you scale it down, the function will never touch the top or bottom edge. From there, use simple algebra to express delta as a function of epsilon. And again, we probably had to restrict our x values to a narrow region around "a", so delta needs to be written as a minimum of that narrow region and the function of epsilon.

[@TimothyLowYK]

Why can't the function touch the top and bottom edges though? Just scaling down a regular rectangle with no boundaries should work just fine to prove it wouldn't it?

[@kingbeauregard]

@@TimothyLowYK I'm just adhering to "| f(x) - L | < epsilon"; notice how it's "

[@youkaihenge5892]

This is similar to the concept of Existence and Uniqueness of solutions. If your solution is unique it must not have any form of overlap inside it's small "neighborhood" at the point, and for it to exist it has a solution. To be both unique and exist implies it is linearly independent and has a "nice" form. These ε,δ proofs can show this concept and for more difficult situations you can use Wronskians. Wronskian is the determinant of functions and its derivatives and if it equals zero you do not have a unique solution and it has dependency somewhere. So, if W≡0 then your solution does not form a basis for your function space. Try this example out with sin(x) and cos(x) to see the beauty of it.

[@tonyhaddad1394]

Wooww good job , youre comment is helpful for deep understanding

[@kingbeauregard]

@@tonyhaddad1394 Thanks! Lord knows I wrestled with it enough; it nearly broke me.

[@mlensenm]

I've always thought that the reason the Epsilon-Delta is presented so early in Calculus I is to scare off those who unprepared. It weeds out many who just aren't ready to take the class. When I took Calc I many, many years ago, things go a lot simpler after slogging through the Epison-Delta problems.

[@stephenbeck7222]

Which is why it should probably be taught later in Calc 1 or just not at all. Maybe it could be saved for Calc 2. Let Calc 1 be a class that focuses on slope and area concepts and applications so people can understand why we even do all this. Honestly I appreciate the conceptual focus of the AP Calc exam over how the standard textbooks are normally used in a college Calc course (and I love how Stewart, Larson, etc. are written generally).

[@imacds]

I think it snuck its way into the curriculum because it's such a short and simple proof to memorize, despite the students probably not even understanding what ∀ or ∃ or proofs even are. So idk if its really that useful, especially if all it does is make kids flunk the class instead of helping them build a basic mathematical intuition.

[@RoyTagliaferro-k2r]

​@@imacds what does not that proof mean

[@TheXenoBrosMC]

wait you do epsilon delta in calc 1?? genuinely shocked (learning it in analysis rn)

[@MyOneFiftiethOfADollar]

Right, a bunch college admins met at Starbucks one morning and thought "what can we do to lower enrollment" ? AND delta epsilon continuity notions immediately raced through their minds!!!! Brilliant insight man.

[@dellta491]

After watching this video several times, I finally understand your proof and also understand your itch to draw a square and shade it. I'm your fan from the Philippines. Excellent work, teacher!

[@GreenMeansGOF]

Small issue with the proof. We also need x greater than or equal to -1/2. Thus it should be δ = min(ε/2, 9/2).

[@blackpenredpen]

Very good point!

[@Sahan_viranga_hettiarachchi]

Yes , its true 👌

[@dijkstra4678]

Where did the -1/2 come from

[@GreenMeansGOF]

@@dijkstra4678 You have to pay attention to the domain of the function. We cannot take the square root of a negative.

[@self8ting]

@@GreenMeansGOF We don't have to do that : when we study limits we do it on the domain, or on the border of the domain. So it's redundant.

[@uvxv_]

As a self-learner, your explanations are mind blowing, thank you sir

[@yassako2496]

HOLD UP, did blackpenredpen just use a blue pen?

[@scottwitoff8932]

The definitions I found confusing and tedious to memorize but once I saw a visual representation the concept was not too difficult. Thanks

[@greghansen38]

I love these videos of yours -- short, focused on a specific problem. Helping me dive back into the math more than a decade after the homework is over. I just wish for an expanded domain, like multivariable, differential equations, linear algebra, all the stuff a physics student would know and love.

[@abhishekkumar4777]

Best explaination of the hardest and elementary topic of Limits i.e. epsilon-delta definination ,I have seen on internet by any teacher😍👍

[@kaylo1680]

If only I had this available when I took calculus when I started my degree. We had an e-d-proof on our exam. Thing haunted my dreams for a good 10 days after said exam. Professors just couldn't explain it in a way that made sense to me. I went back to look at the same problem just now after watching the video and solved it in 5 minutes tops. Damn that felt good. For good measure, it was supposed to be applied on 1/(1+x^2) as x went to 0. I arrived at d=sqrt(ε). I reiterate, damn, that felt good.

[@MiracleMirror117]

Yessssss he finally shaved , thank you for being the best online teacher I ever had, good to see you back to the previous look💖

[@alexj.9011]

Your video is the only video that truly goes into detail on this subject

[@williejohnson5172]

1:15 An absolutely perfect response. You could not have expressed the frustration of trying to explain this con any better. The epsilon-delta proof simply says no matter how close you get to the limit I can get closer. That's it. But the con comes in when they try to convince us that its proof of the limit. It isn't. Conventionally, division by zero is still implied when deriving the limit and the e-d proof has been an inscrutable fig leaf passed around to cover up this flaw.

[@wackeydelly2016]

Finally somebody did an EXAMPLE do show this. I dont know why none of my teachers did this. It pops out so fast with an example and the main idea is pretty simple. Most students get confused because they forget x approaches a limit but will never BE that number and thats why e>0, not equal. Also, even if a function is discontinuous everywhere (see thomae's function) it can still have a limit at a given a value.

[@monkee3613]

I believe in anyone who is trying to learn this subject, I’m 15 years old and I’ve learnt this, to whoever is learning epsilon delta, you’ve got this

[@karlbjorn1831]

I’m 11 and I’ve learnt this! I believe in you!

[@ionikre]

I still remember how I struggled to understand the epsilon delta at my day 1 college life... I believe the reason is that ppl were doing "real" maths before, so it's hard to understand a abstract definition. So a visualized explain would help to get through this. Very good work!

[@navjotsingh2251]

Luckily, I studied computer science and our course wasn't too bad. But, we did have to learn this definition and what helped me was to model it in a programming language like MATLAB and try many different problems, as weird as they can be, see if I can solve the epsilon and delta limit. If not, I'd research why that function didn't work and that's kind of how I got used to it.

[@ainemarkey8359]

Best 20 minutes I’ve spent thank you so much

[@dipun4849]

Thanks sir now I am very happy 😊😊 thanks for your explanation. I am from India and am a small student this was written in my book that delta should be taken as small when you have two values of delta. And that increases my tension. Thanks

[@blakedylanmusic]

I’ve watched video after video on this, and I’ve banged my head against a wall trying to understand it. (For context I just finished Calc 3 and I’m taking my Linear Algebra final) I always wondered why this definition still works with holes in the graph at x=a, and why we write the 0 in 0 < |x-a| < delta. This video answered both of those questions elegantly and now I FINALLY get it! Thank you!

[@blackpenredpen]

Glad to hear 😃

[@citizencj3389]

Once you get into Real Analysis I and II...you will see epsilons and deltas EVERYWHERE.

[@imademedikasurya3917]

after seeing him without his beard, It is like he become 20 years younger

[@aryansudan2239]

thanks bprp. I suck at real analysis and score the lowest in all quizzes. I have challenged myself to become the best in class at it this semester. This is one step forwards in a long journey

[@김태광-j4y]

I am a student who takes analysis in Korea. I understand that example very well. Thank you for your good video.

[@ditang1162]

My professor Dr Grizzle used to tell me that “you tell me how small ε is” when he explained the limit(of series). This is the clearest and easiest explanation I’ve heard.

[@ditang1162]

And yes it was in he’s nonlinear control theory course. You know, Lyapunov.

[@abhishekshukla2570]

Lots of love and respect from India. There is a minor issue of way of talking but I understand ur feelings and concepts also.

[@douglasrauber2040]

I remember when I got my calc book BEFORE my Calculus classes begun, the moment I understood this definition a mathematical tear dropped off my eye

[@julius6678]

such a good-hearted guy, that also happens to be an amazing explainer. Thanks a lot for the video, I took a lot out of it!

[@gabrielfonseca1642]

This video would have been a lifesaver a couple years ago, but even watching it now I have a better sense of what the proof actually says

[@RADARTechie]

This makes more sense than how my course explained it. Thanks! Now to practice this a few dozen times so I retain it....

[@michellauzon4640]

It is very easy to prove the equivalence of the epsilon-delta formulation and the sequences formulation. Which is for all sequences x(n) converging to a the sequence f(x(n)) converges to L. The sequences formulation is very intuitive.

[@bariumselenided5152]

I can’t wait til I finally understand this. My professor did the epsilon delta region thing (he called them “tubes” which I quite liked) , and I get that. Like understanding what delta and epsilon are is easy, and finding one given the other is easy too. But the proofs... I just can’t seem to grasp how it actually is being proven. I can do the work and say the words and get the answer, but I just have no clue what I’m _actually_ doing. Hopefully it makes more sense as my brain marinates in it over the semester. Proofs never were my strong suit anyways, it’s why I wasn’t great at geometry

[@anshumanagrawal346]

I make sense of the definition as: "If x is close enough to a, then f(x) is close enough to f(a)"

[@popularmisconception1]

yes, and including the quantifier part, it says "you can always get f(x) close enough to L, just by getting x close enough to a"

[@anshumanagrawal346]

@@popularmisconception1 Yes

[@herbie_the_hillbillie_goat]

You tell me how close you want f(x) to be to L and I'll tell you how close x needs to be to a. The idea is that there is always some range of values around a that will bring f(x) within any given tolerance to L. This definition solved a long time problem in calculus. While terms like "close to" and "approaches" make intuitive sense, they're vague and don't make mathematical sense. You'll notice that this definition defines the limit entirely in terms of absolute values and inequalities. Thus, we can use mathematics we already know to PROVE the limit is what we claim it is BY DEFINITION.

[@tambuwalmathsclass]

Even if one hates Math, he/she must surely understand this tutorial 👍

[@blackpenredpen]

epsilon or delta?

[@ДенисКосько-н9и]

both

[@orenfivel6247]

if we need to "choose", then: delta.

[@larsbonnet17]

Epsilon, because it isn’t as dangerous as the delta variant.

[@jabahalder7493]

Who invent this definition?

[@MathTutor1]

@@jabahalder7493 Cauchy. The idea was there even in the work of Newton's and Leibniz's, but was not written using ε−δ nations. We call it epsilon-delta rather than delta-epsilon since we choose ε first and then comes δ. Thank you.

[@samarawardasadiya]

OMG THANK YOU SO MUCH, FINALLY I UNDERSTAND IT AND MY LECTURER WON'T BE ANGRY TO ME AGAIN

[@treasure-tf5es]

You are such a great teacher . Thank you so much. I appreciate your effort

[@douglasmagowan2709]

It is definitely the toughest concept in Calc I and is usually taught in the first week. I will say that I didn't really understand it until I was in Real Analsys. I spent about 20 minutes considering the significance of each symbol in the definition before it clicked.

[@sebsplatter914]

Got an exam on this in 2 weeks! Thanks, this was helpful and love your style :)

[@lordstevenson9619]

Same here. Analysis is a struggle and I love bprp’s thought process. Good luck 👍

[@DaMeowster]

Good luck!

[@thebeedy5]

Videos are awesome. First place I come when I'm stuck is here. Great work!

[@simonecp8149]

I'm italian, I understood this better than explained by my analisi 1(calculus 1 name in italy) teacher in my language

[@DaniloSouzaMoraes]

best lesson yet on the subject

[@sandorrclegane2307]

Why is nobody talking about the fact that he shaved his beard! Grats on the new look

[@businesscalculusandbusines8574]

Very nice step by step proof - yes, example is the best approach.

[@B-Mike]

Great!!!! Loved it. First time I really understood the delta-epsilon concept.

[@blackpenredpen]

Glad to hear!!

[@cajintexas7751]

The way I conceptualize it is: L is the limit if, as you get arbitrarily close to x, then f(x) keeps getting closer to L. If every time you "move" a little closer to x without actually reaching x, f(x) always gets closer to L (but never equals or "passes" it), then L is the limit.

[@heartache5742]

you can reach the point and the value if the function is continuous that's the definition of continuity

[@falconhawker]

The Limit concept is beautifully explained by . the sum, as n goes to infinity, of 1/2^n can be shown to equal 1 It is counter intuitive that the sum of an infinite series of fractions is the finite number 1. Limit theory provides a solution to such doubt , IF one understands what BlackPenRedPen> has revealed

[@drpeyam]

Omggggg welcome back 😍😍

[@blackpenredpen]

😆

[@tubeman5987]

I remember epsilon-delta thing being really confusing and I didn't understand it for a long time untill it finally clicked. It was like when I was a kid and said, that telling time by analog clock is hard and I'll never understand it, but then one day I just magically understood it

[@genius5625]

Yes , it is the hardest topic in calculus1. But we have you to make it look so easy . Thank you for all your efforts. 🙏

[@fix5072]

You could also just say that the Funktion is obviously contious so lim f(x)=f(a)

[@jakrispysunshine5844]

I thought the hardest calculus 1 problems were related rates and optimization. Both required a lot of setup and knowledgeable of which method to use since each problem is different from each other.

[@jeffthevomitguy1178]

This is harder in my opinion because it it more abstract and less intuitive for most people.

[@krishnamahawar319]

​@@jeffthevomitguy1178 this is easiest . How is it hard ?

[@jeffthevomitguy1178]

@@krishnamahawar319 delta epsilon is harder than related rates because it requires more thought.

[@treborg777]

It would help to show a counter example for a function limit that doesn’t converge.

[@kingbeauregard]

Obvious example: y = 1/x. There's clearly a singularity at x=0. Less obvious example: y = sin(1/x) at x= 0. Because it's a sine, it can't go to infinity, but there is still no limit.

[@stephenbeck7222]

kingbeauregard For sin(1/x), you can just let epsilon be 0.5. Because of the oscillating nature of the function, you can show that no matter what delta you choose, there is always an x value between 0 and delta such that sin(1/x) is bigger than epsilon - you could choose something like x = (floor(1/delta)+1)/2 * pi, and you’ll get sin(1/x) = 1, if I’m doing the math correct in my head and assuming delta is less than 1.

[@kingbeauregard]

@@stephenbeck7222 Yep. To put it differently, as x approaches zero, sin(1/x) oscillates so rapidly that you can't set delta small enough to capture a single wiggle. So, as you point out, that means you're hosed for epsilon less than 1: it means you can't make delta small enough to exclude the humps (however much of the humps you're trying to exclude).

[@zyad8312]

if it doesn't converge then it wouldn't be epsilon delta definition

[@MADEBYLAC]

just had today mi final exam of calculus 1. I study at spain. That explanation was really good, but i had the luck that i didn’t need to use that definition in my exam

[@nvapisces7011]

Thank you so much for this video! It is so much easier to understand than my lecturer (he's not bad as well but you're better). And the proof is so much shorter. I have to write the part about how you figure out the choose delta. Then write what you wrote. (will ask my lecturer if i could present it your way because he just mentioned that he only requires the part before the proof about getting the delta value in terms of epsilon)

[@michellauzon4640]

It is quite easy to figure the contradiction which is There exist epsilon > 0 such that for all delta > 0, there exist x such that ABS(x-a) < delta and ABS(f(x)-L) > epsilon. If so, we can put delta = 1 / n for all positive integer, pick x(n) for each n. Then X(n) tends to a, and f(x(n)) doesn't tend toward L.

[@philj9594]

Okay I feel a little bit validated now as I dropped my calculus 1 course this summer partly because when I got to the epsilon-delta part of the textbook it made me feel like an absolute idiot completely out of his depth. Everything outside of that was coming to me fairly painlessly (struggled a little with related rates). Going to give it another try this fall. Thank you!

[@Omar-i7y6w]

thank you god for putting this video and in my recommendations and thank you professor for making this video

[@dafta31]

I studied the definition using neighborhoods from "Vectorial Calculus" (Marsden, Tromba) and it helped me a lot. Please do an example proof when the limit does not exist.

[@jimbyers3092]

Very clearly stated process to help one learn to prove limits! Thank you.

[@beginneratstuff]

I wrote the definition in words to aid my own understanding. Maybe it will help someone: The _lim x->a f(x) = L_ if for all distances _ε > 0_ away from _L_ , there exists a distance _δ > 0_ away from _a_ such that when the distance between _x_ and _a_ is less than _δ_ , the distance between _f(x)_ and _L_ is less than _ε_ . In other words, if I can choose an arbitrarily small value of _ε_ , and I can always find a value _δ_ (based on that value _ε_ ) so that when I choose x-values that are less than _δ_ units away from _a_ , the function outputs y-values that are less than _ε_ units away from _L_ , then the limit as _x_ goes to _a_ of _f(x)_ exists and is equal to _L_ . Also note that the distance between _x_ and _a_ should not be 0, since the definition of a limit does not consider _x = a_ .

[@karlbjorn1831]

Helped me! But I’m still struggling to understand this

[@HeroicVigilant]

I'm actually in a course right now that is all about limits and infinite series, so this couldn't have come at a better time. I have to do so many limit proofs and I didn't really understand what was going on until now, thanks!

[@lordstevenson9619]

Would love if you could do more ε-δ definitions, maybe from sequences, series, continuity and such. I’m finding your explanation and thought process very helpful, especially with how to go about doing the proof and how to formulate a valid proof. I’ve got exams on Analysis in a couple weeks so hoping it goes well. 👍

[@citizencj3389]

Its all the same definitions just dofferent arrangement of quantifiers.

[@OmerAgmon]

Great video as always. I especially enjoyed seeing you moving the mic away (5:43) for what might be the first time

[@blackpenredpen]

😆

[@orenfivel6247]

i think we can choose δ=min{|(3-ε)^2-9|,|(3+ε)^2-9|}/2 as a tighter choice for δ, basically via solving for x-4 in 3-ε

[@blackbeanboi]

This video really helped me. You are a great instructor and I really appreciate your way of teaching! Also I love the pokeball, kirby in the backround and the flannel! I love all of those haha!

[@popularmisconception1]

It becomes clearer if you name epsilon and delta meaningfully instead of just giving them obscure letters as a name. Given epsilon is an output space error margin (or image space or codomain error margin), choose delta to be input space approachment margin (or source space or domain or whatever words you think in about functions). Now you can say that L is a limit of f(x) for x approaching a if FOR EVERY output space error margin THERE IS an input space approachment margin SUCH THAT if you approached the input of the function within that margin, you also made an error on output within the desired margin. So it means that YOU CAN GET the value L from f(x) AS CLOSELY AS YOU WANT just by approaching your intput x to the value a sufficiently. As closely as you want. Always. Really. Just go close enough to a. That's what the epsilon delta says in human words.

[@popularmisconception1]

Which makes me think, you could have a relaxed version of limit, let's call it an interval limit for now. An interval limit of a function f(x) for x approaching a would be an interval [L,U] such that by approaching x sufficiently, you can get f(x) into [L-epsilon, U+epsilon], and you would say there is no limit interval if one or both of L, U is infinite or too large or something like that. With such a limit you could have an almost continous function :D Just small finite bumps here and there. That could be handy sometimes maybe.

[@TheLycheegreentea]

Delta and epsilon proofs isn't exactly everybody's cup of tea. When I did it back in High School for Calc, it barely made any sense to me. Once I started Real Analysis in college, then it became better. But I still noticed that the computation is very limited due to the fact that you have to work backwards to figure out what you need to "choose for delta". That's why majority of these problems are polynomials or square root functions. The numerical approach is easier for all students to see and understand (most of which that aren't going to major in math). The proof part will always be what I call "math-terbation". At the end of the day, it just makes you feel better but nobody else. -_-

[@GarryBurgess]

Because I understand this, in my retirement years, when I haven't studied math for decades, I'm going to reward myself with a Dairy Queen Blizzard.

[@SyberMath]

Very nice explanation! Great proof! 👍😍

[@blackpenredpen]

Thanks!!

[@noahali-origamiandmore2050]

What if since sqrt(2x+1) is always either positive or 0 and 3 >= 1, I just ignored that part and said that

[@tobybartels8426]

Yes, and this is probably how I'd do it, since you get a larger (but still valid) δ for the same amount of work.

[@stephenc7970]

i just want to say that epsilon delta might be the most useless thing i’ve ever learned in my life

[@bestopinion9257]

That's the coolest part in calculus.

[@richhamster24]

you are a brilliant teacher thank u very much

[@DergaZuul]

As for me this explanation makes more complicated than it actually is. All you have to do is to find delta(eps) function so that inequality holds. In some sophisticated examples of limit you only can show existence of delta but not its actual value, it is still sufficient.

[@roberttelarket4934]

Again as I commented to you one month or so ago for the epsilon/delta definition the wording is critical and I was penalized in the advanced Calculus exam I was given in 1969. It is NOT for ALL epsilon, but for EACH "small" epsilon there is a corresponding delta. . . . .!!!!!

[@blackpenredpen]

It should be for All, that’s why we have the upside down A.

[@roberttelarket4934]

@@blackpenredpen: I’m not going to argue again as I did several months ago. The upside down A or any other symbol can be used for anything but here it must be read FOR EACH and then has to be followed by CORRESPONDING! Again keep in mind this is as an undergraduate in 1969 correcting me on an exam by a full professor of mathematics!!!

[@billh17]

Robert Telarket said "but for EACH "small" epsilon there is a corresponding delta..." You need to show the condition is true for each epsilon even "large" epsilon. That is what "each" means. Of course, you need to mention that epsilon must be greater than zero also.

[@roberttelarket4934]

@@billh17: No! Not for large epsilon. Only for small epsilons greater than zero even though small is a relative and sort of an ambiguous, vague amount but is understood by those who understand the concept/definition!!! Once again the capitalized words are critical: for EACH epsilon greater than zero there is a CORRESPONDING delta. ALL can be misinterpreted as for all epsilons greater than zero there is ONE delta which is wrong!!! To repeat I lost 25 points in 1969 on an advanced Calculus exam for that mistake-graded by a full professor mathematics!!! I will not discuss it any further! Thank you.

[@billh17]

​@@roberttelarket4934 said "Not for large epsilon." If I want to prove the statement "For all epsilon > 0, epsilon * epsilon + 1 > 0" then I need to prove it for each epsilon > 0, not just for small epsilon. @Robert Telarket said "ALL can be misinterpreted as for all epsilons greater than zero there is ONE delta which is wrong!!!" Yes, that is true (since English is ambiguous). That is why it is usually stated that the delta depends upon the epsilon (although this is really not needed to be stated since "for all" is a technical term in first order logic where the rules of inference make clear that "(Ae)(Ed)(...)" is different from "(Ed)(Ae)(...)" where the latter is the one which states that "for all epsilons greater than zero there is ONE delta"). English is ambiguous in using "all". For example, consider the statement: "All people have a soulmate." This does not mean there is one person that is the soulmate for all people. Again, "for all" is a technical term in first order logic. I agree that the informal English usage can cause confusion (as you point out). One could use "for each" as you point out or one could use "for every". These variations are probably not used because they both begin with "E" which conflicts with the "E" which stands for "there exists". Also, there is an historical reason to use "For all" since Aristotle in his development of logic used "All" (the classic example being: All men are mortal, Socrates is a man, therefore Socrates is mortal.) @Robert Telarket said " though small is a relative and sort of an ambiguous, vague amount but is understood by those who understand the concept/definition!!!" But the statement should not be sort of ambiguous and have an implicit understanding of what is meant. That is why there is no restriction on epsilon being small since it is not needed in the precise definition for "limit" (which is stated in the video and does not require epsilon to be small). If epsilon were required to be small, then the definition would need to specify that. Otherwise, the definition is not a correct definition since the requirement of being small is not explicitly stated.

[@sapigembel]

Instead of

[@peterg76yt]

Where the formal definition appears counter-intuitive is it starts with epsilon implies something about delta and then ends with delta implies something about epsilon and also the definition doesn't explicitly say anything about arbitrarily close to zero even though that's the important part. It becomes clear when you picture the moving parts and appreciate how epsilon close to zero implies delta close to zero.

[@davesmyers]

Around 9:20 you might discuss more deeply that sqrt(2x+1) does not give a symmetric window for delta, and you might indicate which is a better choice for summarizing the inequality: 0.58 vs. 0.62. I know this does not end up mattering in the end, but it is interesting perhaps and then you can mention how it does not matter when eps/2 is so small by comparison. Just a thought.

[@alldabbing8930]

BPRP Single handedly saved all of us

[@mrm6696]

I love your vids man. I like to take some time off of my extra math studies on the weekends and these videos are a great way to relax and enjoy some math.

[@mesganawoldeselassie4267]

I finally understand the definition thank you bro

[@kushsachdeva9340]

Finally a good example… WHY CANT CALC PROFESSORS TEACH LIKE THIS!!!

[@shivrajpatil1770]

I feel like I can conquer entire calculus class now.

[@michapodlaszuk9025]

Knowing this stuff makes me feel a lot cooler B)

[@russchadwell]

Yeah, this one nearly killed my ambition. But, it cannot! It's really important!

[@MyOneFiftiethOfADollar]

An old calculus textbook by Salas chose express delta in functional notation fashion. Delta as function of Epsilon. Delta(Epsilon)

[@RahulGunwani]

Thank you Sir, your explanation is really helpful 🙂

[@ahcenecanpos9463]

nice and logical explication .............. very good

[@flobah]

This helped me understand a lil more! Thank you!

[@الرياضياتالعظيمة-ز6ك]

The definition at first time is so difficult in the meaning and relation to our understanding to the limit concept and also in the calculation or its application. You can also talk about the Riemann integral definition.