Epsilon-delta limit definition 1 | Limits | Differential Calculus | Khan Academy

[@zualapips1638]

Published in 2009 and saving my life in 2020. Thank you so much!

[@descending_misery9018]

Haha same..

[@heisenberg7945]

21/22

[@JemalHamid-e2m]

Me 2024

[@STUDYWITHPIYUS]

​@@JemalHamid-e2mMe too

[@random_human_1879]

2025 🥳

[@Sarah-vi8px]

I'm in Calc 1 at a university... I'm learning more watching Khan Academy videos than being in class.

[@bik8353]

saame.. my teacher is chinese and speaks at a pace of bullet train

[@misafan4u]

I stopped going to my lectures because of this lol... I'm 2 weeks ahead of my class.

[@RajanJS]

Really, In India they try to ram this down the throats of 16-17 yrs old. Best part is that some of tose kids actually score 98-100% in their 12th Class Boards and this is just the basics. PS. I studied this 30+ yrs back and am thankful to Mr Khan for making it so interesting. Allah give us more Mr Khans who spend their energy and time making these videos.

[@babyonabilify]

Are you kidding me? This is some sloppy-ass presentation. Where is Khan himself? Outsourcing his "educational" videos is pretty stupid and leads to poor and poorer quality. Just glancing at my professor's notes is 10X more helpful than this etch a sketch bullshit.

[@uahfiuahflhvli]

^^ then why are you here? go study your notes

[@kabio007]

why aren't lectures as informative as this?? just goes to show a lecturer may have a PHD in maths but no clue how too teach. This video deserves an award ..welldone!!

[@DirisalaSandeepKumar]

yes

[@davidlozinski9665]

When the guy on youtube cares more than your actual instructor does about your mark.

[@shadmankhan6041]

Lol... good one.

[@mkutkarsh]

Let's teach for mastery, not test scores - Sal

[@leokeatonn]

God I'm so glad I was born in the 2000s, I have no idea what I would've done if I didnt have instant access to wonderful people like Sal

[@Deescizzle]

We asked a lot of questions in class and really bothered our instructors. And, we only took Calculus for the best instructor. Otherwise, we'd have to retake the class, not fun

[@davidb2416]

I used Schaums Outlines, they were no where near as good as Khan Academy.

[@popsicIes]

@@Deescizzle Do you think it was more beneficial since everyone was asking questions?

[@Deescizzle]

@popsicIes I think so. I like the interaction with a person. I feel that I definitely learned more, and it stuck in my head better. I can learn online and watch videos, but it doesn't have the same lasting effect.

[@khanacademy]

You are correct. The definition does not help you to find the limit. You need it when you want to prove that a value really is the limit (a value that you would determine using other means).

[@Mattmath]

When I first time heard about "Epsilon Delta", I thought it was some sort of RPG game boss.

[@v3xt0n]

HAHAHASHFLWELR

[@mohandas3212]

Ya mee too 😂

[@NugisBiboSchicken]

deltalovania

[@vivekkorupolu4996]

When First I heard epsilon The thought which came into my mind is the GTAV epsilon

[@khanacademy]

I've been meaning to do a video on this for a while and forgot about it. Your email was indeed the catalyst.

[@nickpickle]

its ok bro

[@nada3131]

Psionic reassuring him 11 years later, 😂, love it

[@azyle2104]

@@nada3131 lmaooooo fml

[@brennancoleman3682]

I think everyone here is doing some panic studying so we don’t implode next year

[@blackpepper9828]

vu88u

[@julesvg3111]

Man I really needed someone without a heavy dutch accent to explain how this works, in a way that I don't fall asleep And this video is JUST what I needed, thanks!

[@chloemckinster320]

When my calculus professor "explains" this i swear she is speaking a different language. You're a life saver Khan.

[@straystars2492]

Cheers to this video for uniting troubled calc students for over a decade.

[@Plamen2007]

you are the homiest of homies..

[@prathameshsawant5574]

what is homies?

[@Chapz725]

@@prathameshsawant5574 basically its another way of calling somebody your friend

[@MelTurpin]

Excellent video! I think the thing that makes this so difficult for students is the number of variables involved in the formal statement. Look at those last 2 equations you wrote: 0 < I x -a I

[@allanwei9709]

10 years later this video still better than the lecture I got in class...

[@firebelias1266]

I got this in my third week.....And we have to study it ourselves Oh God thanks there is Khan academy.

[@iPinkheart]

I cannot thank you enough, Khanacademy! I am currently enrolled in an online calculus course, and I have seriously no idea what is going on until I remember your website! Before coming to this video, i starred at the page on the textbook that talks about this epsilon delta limit thing for almost an hour, not understanding anything.... I spent this 12 minutes and 48 seconds way more effectively than the hour I wasted!

[@cxp.b]

so i will try to get you 13 yrs back ( if you still alive ) . how are you

[@ouizaouiouiz]

thanks!!!! that's the kind of explanation I've been looking for for the past two hours!!! thank you so much!!

[@Daski69]

I've watched a bunch of videos about the topic and it's crystal clear to me that if the limit of f(x) is within epsilon of L, then x is within delta of a. I can visualize it, express it mathematically and it makes perfect sense, however, how does that prove a limit???

[@TherealLaserdog]

it don't because calculus book is dumb

[@秦海-r5l]

the limit, in this video, means every time you pick a x in the range of (x-delta,x+delta), the f(x) will definitely falls into the range of (y-epsilon,y+epsilon). Thus the limit is epislon, which can be as small as u want.

[@wdavis6814]

It'll make more sense when you start taking derivatives.

[@Deuce1042]

If you understand that then I don't get the confusion. If for every epsilon greater than 0 there exists a delta greater than zero such that the conditions involving the absolute value inequalities is true then the limit exists and is equal to whatever you were trying to prove that it was equal to. Epsilon can arbitrarily small in that case, and the absolute value of the difference between the function and it's limit will always be smaller than that epsilon.

[@Deuce1042]

Think of it this way. Given some x value between a (which is the x value you're approaching) plus delta and a minus delta, if the limit exists and is L, then that guarantees that the function f(x) lies between L plus epsilon and L minus epsilon. Now I give you some epsilon greater than zero, it can be even infinitesimally small so long as it isn't zero. Then that means, if the limit is L, you can give me a delta greater than zero such is between L plus and minus the epsilon that I gave you. Now do you see it?

[@Magoogan92]

If you aren't already, become a lecturer. You explained this in such a simple way that I understood what the hell was the point in delta and epsilon in 5.5 mins as opposed to sitting in hour long lectures wondering what's going on! Thankyou!

[@andybraamawitz9395]

Sal Khan - One of best people on the internets, period. He explains it far, far better than my $120,000 professor ever could.

[@peterthegreat996]

so essentially calculus is simple algebriac inequalities, functions, factoring, and arithmatic with greek letters.

[@BJ-no2oe]

Mr Petester z Basically, Simple Aritmetic: Logic Algebra: Logic Geometry: Logic+memorization Trig: Logic Calc: Logic Math is easy if you use logic and know how the rules work

[@理科角度看历史]

Actually ,it likes a magic ,right?

[@TechToppers]

Calculus does not exist. *Math* Exists.😇😇 Edit:- The point is... *Common Sense* exists.

[@Sk8erMorris]

@@BJ-no2oe Geometry is not memorization... it's learning a definition and understanding it.

[@好吧-h6k]

@@BJ-no2oe I mean all of them are memorization, just geometry is more difficult to remember

[@metsfan12694]

Khan academy teaches me more about Calculus than my professor does. The visuals in these videos help me so much. Keep doing your thing, Sal!

[@iqrafatima5549]

What do you do now?

[@56Valo]

lol, "included in the third week," he says... First week, here. >.> Thanks, Khan Academy.

[@pointman6778]

Same

[@Triplechoco52]

Valo56 2nd week here

[@Noone-ps6tl]

Same

[@haileywarner5109]

nice profile comrade

[@rachelll792]

Ikr , this was taught in the first day of engineering math class.. Wow thank god for khan academy!

[@edensalvia]

Bless your soul. Right at 11:26 it all clicked and I have been enlightened. THANK YOU!!

[@leroyjenkins8886]

I learned this in the second week of calculus i am so ready to drop out but I can't

[@gravitywaves123]

I am going to nominate you for a special noble prize for mathematics.Great stuff. Keep up the good work.

[@cassied9327]

At 4:55 Sal is pretty much like, “We’ll talk about epsilon and delta later, nvm we’re taking about them now” 😂😂

[@yiya1717]

OMFG! DUDE YOU ARE A SAINT! I WOULD HAVE FAILED MY EXAM IF IT WASNT FOR YOU! Bcuz my instructor doesn't speak english well, and I don't know what the hell she tries to say during lecture.

[@Zander101084]

this is pretty easy if you just sit down and pay attention to it. YOU'RE AWSOME SAL!!!

[@KleinGrass]

English is not my native language, but i understand you way more than i understand my teacher, who speaks my native language. Thanks :D

[@olyayatskiv3547]

I can't believe that I'm watching this on a Saturday night just for fun :)

[@chancehackenbruch5529]

Still helping students in September 2021, thank you!

[@jackstorm777]

I read this in my textbook and was COMPLETELY confused, you have totally cleared it up for me! YOU ARE AWESOME!!!

[@KirkSH52]

This would have been so beneficial if Kahn would have used a marker board or a chalk board. This computer program is really sloppy and messy. I would love to see all these video lessons where he writes real neat on a marker board. He is very intelligent and a great teacher but the writing is far too messy and sometimes it cannot be interpreted.

[@3lderz]

this is one of his older videos. I think he used his mouse for this one, but now he uses a pen tablet for his newer videos

[@KirkSH52]

3lderz That would be a lot better than is big mess.

[@DubStepMTL]

You should probably wear some glasses if you couldn't read what was written. Especially since he describes what he writes.

[@KirkSH52]

DubStepMTL He is a messy hack.

[@KirkSH52]

Hacky Khan.

[@emcue786]

thanks so much! huge test tommorow and my professor and TA and tutors did not help at all!

[@hanthonypiano]

Writing everything down along with him helps even more... this guy is great

[@radashevskiy]

omg finally someone explained this so that it actually makes sense. the key phrase for me was the " Ican guarantee you that..." Thanks so much! now i dont like my calc teacher even more

[@vishwanathsharma73]

OMG! I've watched so many videos and read books but didnt get this concept, thanks for explaining this in a easy way with graphs🛐

[@discuss2130]

NIce graphical illustration that explains the ideas behind the epsilon-delta limit definition.

[@kevincrawford6864]

Got more from a 12 minute khan academy video than 1 hour of reading in the book, thank you!

[@guranshsingh2868]

Thanks alot Khan Academy for this video.

[@Cagebreaker21]

never did this prior college.. needed to watch twice. thanks good video

[@implicit2656]

Watching this 8 years later than today... Still extremely helpful and intuitive

[@suvalakshmi2874]

Me 11 years🙋🏻‍♂️🙋🏻‍♂️

[@nabeelhisham3774]

14 years

[@Kate-wz9vx]

Published in 2009 and saving my uni life in 2021. Thank you so much! XD

[@pranagy]

12 minute video = what 3 hours of lecture time couldn't explain. This guy is amazing

[@DirisalaSandeepKumar]

yes u are correct.did u passed ur calculus exam all the best

[@Heisenberg_1_0_0]

The most important part of this video which made me crystal clear about this topic is "I CAN GUARANTEE YOU"...

[@brilliantfranz]

excelent, i never really got to actually understand this thing even though, I did a lot of exercises findind limits Cheers from Venezuela

[@ad2181]

Sal, Since you must know the "limit" before you apply the epsilon-delta technique to prove the assumed "limit" is true. Epsilon-delta is only a test not a limit finder. Please show some counter-examples, For example (x->3) of (X^2)=10 or some clever function that would be on a standard test for the purpose of proving some people's understanding of the concept a less than others. Thanks, P.S. I've enjoyed your financial-economic series, your anaylsis is excellent.

[@Saypayaa]

This is so awesome! You should get a Nobel Peace Price for preventing so many students from ripping their own hair off. Good job and thanks a lot!

[@anticorncob6]

Just watched the rest of the video. It seems everything I said is right except that instead of |x - c| < d it should be 0 < |x - c| < d because x can't equal c. Oh well. Thanks a million for this, I've tried to understand it on my own, but nothing really worked until now.

[@wiky8288]

I'm so enjoyable today's third week calculus lecture after watching this video a night before the lecture! I have a mixed feeling that why the lecturer tried so head to explain the basic concept in an hour and making people confused!

[@daniloalvaro2142]

This is the best video I watched within this topic

[@kevinparsley6806]

thank god for resources like this. i remember when people that were good at math kept all the information secret.

[@rambo1525298]

thx a lot !!!! if all math teachers were like you all the people in the world would be mathematicians.

[@tunamay0168]

Thank you for making content that helps us students with concise and thorough explanation ! All Heroes don't wear capes !

[@larrypatterson3957]

I’m nearly 73 and 9 years ago I took my first Calculus I class at Palomar college in CA. Prof. David Lowenkron had us remember this. “ Let f be a function defined at every number on some open interval containing C, except possibly at the number C itself. The limit of f(x) as X approaches C is L, if, and only if, given any epsilon >0, however small, there exists a delta >0, such that 0< I X-C) I is < delta then I f(x) -L I is less than epsilon. Which means, you can take f(x) as close as you want to L by taking x sufficiently close to C, without touching C” or words to that effect. Now……where are my car keys?

[@MisterBinx]

This tripped me up and I'm in multivariable calculus right now. I'm a couple sections behind because I spent too much time on this part with two variables. I really wish this would have come later.

[@krpcannon123]

Isn't the pedagogy in mathematics so frustrating? They teach things in such a backwards order, thinking they are making it easier on us, but really it just trails us along with convoluted thinking...

[@brianm.4368]

Very interesting. Can you elaborate?

[@unnikrishnanvalappil6973]

Isn't it like learning about the skeleton before learning parts of the body in play school ?

[@Tj-wn3qg]

I use these videos to fall asleep

[@NA-gx9cr]

A Khan Academy Classic.

[@starlinej]

I never learned this in Calc 1, but we just started learning about it in Real Analysis and this video was a big help.

[@harjitsingh7308]

Yeah this isn't a calculus topic, it leans more towars real analysis.

[@albertwesker2050]

Hey do you think you could update this video? I find it less intuitive than the other videos. I understand it, but this video needs updated compared to the others. However, I am eternally thankful for K.A.

[@aurelysmarie7417]

when you're taking calculus over the summer, and he says they teach you this over the first three weeks and my professor gave it to me the first day :)

[@RealEverythingComputers]

Awesome video! Even after 15 years! Thanks

[@marylamb6063]

It's best to be very familiar with the language of inequalities to prove without a doubt that a limit exists for a function. We learn about inequalities early in algebra but by the time calculus rolls around we have forgotten inequalities. This is a good video to get an intuitive idea of a what a limit means, but the algebra of inequalities must be used to prove "without a doubt."

[@erikdahlen9140]

In multi variable calc and need to revisit this definition, very helpful video thank you!!

[@Mathaniac]

don't know why the concept was so difficult to understand at first but i finally got it. thanks!

[@wingegia]

wow, great explanation of epsilon-delta theorem. you explain better than most teachers. cheers, im glad i watch this vid.

[@dhanmaysingh5843]

Thnx to khan academy ...Finally I got it..😍

[@Gothorochimaru]

Khan Academy makes Calculus fun for me!

[@5gallonsofwater495]

Oh. My. Gulay. This is very helpful, thank you Sir

[@anticorncob6]

f(x) = (2x^2 - 2x)/(x - 1) seems to graph what you want. That was easy to figure out, but the function was irrelevant, now let me try to come up with the definition on my own. If limt x -> c of f(x) = L, then for any E > 0, there exists a d > 0 such that f(x) in the range of c - d to c + d will always be in the range of L - E to L + E. Or, any x such that |x - c| < d, then |f(x) - L| < E. Is this correct?

[@avilesnba]

oh man great stuff....kudos i just started my first ever course on calc....and the whole limits thing was confusing. Like you said, i was confused about this stuff and we were already moving into derivatives.... thanks for the help though, i completely get it now

[@joshuayap2005]

Now, i understand those delta-epsilon things. Thanks very much!

[@1986rahel]

the best math teacher ever!!!

[@odeforodds]

Thought this was hard, not anymore. Thank you so much!

[@NickPlante]

Thank you Thank you Thank you! This saved me for my test tommorrow.!

[@baarbie0786]

OMG thank you so much!! I was so confused in class and this really helped me understand the definition!

[@sameeoh]

thank you sooooo much. this cleared any questions i had with this definition. your my calc savior.

[@rachelll792]

Thank you khan academy ❤

[@amroelkhodrai3048]

old but gold

[@TheStfu1000]

listening to some lecture for an hour with someone maths person who's numbers and writing is all unreadable is an ancient way to learn. really one above the age of 13 could get a well rounded education from wikipedia and the internet.

[@vishalshinde5252]

I thought I understood math until I started studying Analysis

[@briancaine4789]

The third week? My professor covered all of this and the next video on the FIRST DAY of calc I

[@nstsgr]

Thnx so much for this video. Before seeing that i was in great loss.. like i studied 40 pages and didn't understand enough

[@DrBlokmeister]

Thank you! This made calculus a lot clearer.

[@siddharth93214576]

Execellent!! You explain really well. Finally - a rigorous mathematical definition of the limiting operation which connects calculus to our more familiar mathematics

[@BabikerAlbanna]

9:58, Should it be a range around the limit L of f(x), or as said range around f(x)?

[@Jean-qu1mf]

does anyone know what application do they use to draw the graphs?

[@recepcoban7770]

mathlab or geogebra

[@VanHalen2882]

this video gave me so much clarity, thanks for teaching so down-to-earthly!

[@Sacchidananda]

actually superb ,conception cleared😄😎

[@aeroboi2862]

Thank god you exist

[@mrcrazymadwhat]

over simplifying it the if you have the equation y=x and you want to get close to x=5 so that is 5,5 you want to be within 1unit of that point on the y axis then want would be that range of point on the x axis. so you on the y axis you want to be between 4-6 so to get those you need to have x the value between 4-6 sal did i summarize the idea correctly???

[@DzSchuur]

Thank you very much, never got it with calculus. Now I do!

[@chriscampanelli3769]

Thank you for everything, Sal

[@mindy3093]

Best explanation ever 😃😃

[@Vorbis5]

And this year of Calc AB will be a breeze. TY.

[@scamander24]

Thank you so much!!! You've helped me more than my math teachers lol

[@pithikoulis]

@xjaat you mean epsilon and delta? because if they are 0 then it means that they are the same as L and a. That's something that we don't want. We want values that are close to but don't touch a and L

[@cxp.b]

thank you man

[@Okarin-gq6ky]

understanding with a diagram is always better ...