# Understanding epsilon-delta proofs

First, get and read this. Really good.

## before you even START with epsilon-delta . . .

| x – 3 | < 1;

SAYS in English “x is within 1 unit of 3.”

You have to be fluent with that idea before starting ε-delta; proofs.

Let’s look at a picture of this inequality on the real number line:

So for example, if x = 2.8:

| 2.8 - 3 | < 1 | -0.2 | < 1 0.2 < 1 === TRUE

Now let’s try for x = 3.5.

| 3.5 - 3 | < 1 | 0.5 | < 1 0.5 < 1 === TRUE

So, | x – 3 | < 1 MEANS *x must be within 1 unit of 3 on the real number line*.

To be able to understand epsilon-delta at all, you have to get into the habit of looking at an inequality like

| x – 10 | < 5

And immediately just say in a snap “That inequality says that x is within 5 units of 10″

Or

| x – 0.4 | < 0.00001

And say “That inequality says that x is within 0.00001 units of 0.4.”

In general, for any inequality with the format:

| x – c | < a

That says in plain english that “x is within a units of c.”

Epsilon-delta proofs are *actually* easy.

## The **meat** of epsilon-delta proofs

The meat of epsilon-delta proofs is just this idea.

if

0 < | x – c | < δthen

| f(x) – L | < ε

## Epsilon-delta says:

AS WE RESTRICT x to being within δ units of c, then, as a result of that restriction, f(x) becomes restricted to being within ε units of L.

If the above statement is true, then and only then can we say

lim f(x) = L x->c

So, here’s the “definition of a limit”, but with more explanation in English words:

The limit:

lim f(x) = L

x->c

existsif and only ifwhen we restrict x to be within δ units of c,

0 < | x – c | < δ

then, as a consequence of that restriction, we in effect are restricting f(x) to be within ε units of L.

| f(x) – L | < ε

If that happens, then we know that the limit of f(x) as x -> c

equal to L.is

## Ok, but how do you do an epsilon-delta proof?

So here’s an example of how this stuff works.

Use epsilon-delta to "show" that lim 3x - 3 = 12 x->5

I know what you’re thinking. “Can’t we just do this:”

lim 3x - 3 x->5 = 15 - 3 = 12

but nooooooo! That’s not good enough for epsilon-stupid. You must “show” it.

Use epsilon-delta to "show" that lim 3x - 3 = 12 x->5

So, what we use the “definition of a limit” as stated at the top of this page (you should memorize it really for use on tests (YES they DO ALWAYS put epsilon-delta on tests . . )

The limit above exists if and only if for each ε > 0, there exists a δ > 0 such that:

IF0 < | x - c | < δ [ c = 5 though, plug in: ] 0 < | x - 5 | < δTHEN| f(x) - L | < ε [ f(x)= 3x - 3, and L = 12, plug in: ] | ( 3x - 3 ) - 12 | < ε

So read that in English as:

“IF x is within δ units of 5 . . . “

“. . . THEN ( 3x – 3 ) is within ε units of 12.”

The key to epsilon-delta proofs is you have to **relate epsilon and delta**.

You have to argue that IF 0 < | x – 5 | < δ ( x is within δ units of 5 ), THEN we can conclude that | ( 3x – 3 ) – 12 | < ε (THEN ( 3x – 3 ) is within ε units of 12 ).

HMM! Hopefully, this is starting to make some sense. Here is how you proceed.

Let’s work with the THEN part (the | f(x) – L | statement), and break it down a bit:

| ( 3x - 3 ) - 12 | < ε | 3x - 15 | < εLet’s FACTOR (because we love to factor)

| (3)(x - 5) | < ε 3| (x - 5) | < ε | (x - 5) | < ε/3;

That looks familiar! Suddenly, the | f(x) – L | < ε looks a lot like the | x – c | < δ statement.

HMM!!!

We can ** relate** the epsilon statement and the delta statement ( 0 < | x – c | < δ ) in this way:

CHOOSE δ = ε/3. (Get used to the idea of “CHOOSING” δ)

Then, we go:

| x - 5 | < ε/3 [ CHOOSE δ = ε/3 ] | x - 5 | < δ

WOW!!!!! How marvellous. It will seem very very strange to you that in the middle of this “mathematical rigor”, we end up going and “choosing” δ = ε/3. You’ll see that this “choice” doesn’t really hurt the “rigor” of what we’re doing though. . . just keep at it.

Next we have to show that this “δ” we’ve chosen ( δ = ε/3 ) “WORKS”.

So we go back to the original statement:

IF0 < | x - c | < δ [ c = 5, chose δ = ε/3 ] 0 < | x - 5 | < ε/3;THEN| f(x) - L | < ε [ f(x)= 3x - 3, and L = 12, plug in: ] | ( 3x - 3 ) - 12 | < ε | 3x - 15 | < ε 3| x - 5 | < ε | x - 5 | < ε/3

Wonder of wonders! It “WORKS”, because the statement has now changed from:

“IF x is within δ units of 5 . . . THEN ( 3x – 3 ) is within ε units of 12.”

To:

“IF x is within ε/3 units of 5, THEN x is within ε/3 units of 5.”

Which cannot be argued against.

Remember, its not stupid. Its “rigorous”.

## 125 Comments

holly crap i love you…if your a girl can i marry you? no like seriously, i mean it – I. LOVE. YOU.

glad it helped!

WOW. i’ve been trying to figure this out all day. and now I suddenly understand within 5 mins. You’re a life saver. :)

THANK YOU SO MUCH! i swear calc professors are useless. and the whole epsilon-delta thing is somewhat useless, its so repetitive. anyways, u mind explaining the way to prove the limits of crazy polynomials? by using those given limit laws , they are so confusing!

and when the number infront of |x-c| doesnt work out to be the same as it shows up with delta, we were told to let it be smaller than a constant and find the constant. that is confusing too. do u know what im saying? lol cus im confused about what ive just mentioned too . . .

It’s beautiful (it really is) how well you’ve explained this concept. Well-done, and thank you.

dude you are my god! Thanks soo much, I’m going to give my idiot teacher an anonymous note to look at this site!

Thanks for the lucid explanations. Makes everything much more clear now. Probably some book authors should read this and take tips.

Brilliant!@#$% That’s an amazingly clear Explanation. You just explained what 5 Calc lectures didn’t teach me in about 1.7% of the time. Good work.

Dont delta-epsilon proofs require you to show that as delta decreases, so does epsilon? If epsilon increases while delta decreases, the limit doesnt exist… right?

Im still struggling with this concept.

See, every demonstration I have seen, the tutor picks and chooses a nice, simple linear function ahead of time. What if you used f(x) = x^2. Or f(x) = sin(x). Or f(x) = x/x as x -> 0. Your example work isnt so apparent or intuitive any more.

How do you use d-e to prove sin(x)/x, as x->0 equals 1?

See… you have to already know your L beforehand. You use d-e to DEMONSTRATE L is the correctly presumed limit, not to prove L is the limit. Right?

Because you have to know L first in order to use this. In fact, you have to know what x approaches first in order to assert what the limit approaches… you have to know your x->a and your f(x)->L.

But what if you pick the wrong L? What if L isnt your limit? How does d-e fail?

I think the point of e-d proofs (key word being proof) is to prove the L, after you already know it.

Sweet explanation though, helped a lot.

Aloha CogitoErgoCogitoSum,

I agree with you that these epsilon-delta proofs are not proofs at all the way they are usually done. They have always disturbed me because the logic is circular and very unsatisfying. That last step of “showing that the delta works” is really only checking that you did the correct algebraic manipulations the first time around (or are consistent in making the same mistake). It’s like verifying your solution to an algebraic equation. You are also correct that nearly every time someone does one of these “proofs,” they pick some very simple linear function, but more important, they never show what the “failure mode” is for trying to prove the wrong limit. In fact, there is no “failure” (at least none corresponding to anything stated in the e-d definition) if you follow the steps above using the wrong limit. Try L = 13 instead of 12, for example. With proper algebraic manipulation, you get delta = (1-epsilon)/3. The only problem I see with this delta is that when epsilon goes to zero (as you would want), delta goes to 1/3 (which is absurd), but NOTHING in the e-d definition says this is a problem. Do we have Cauchy to thank for that, or have the calculus textbooks been remiss in not stating this requirement? I think you stated it correctly when you said that there is ANOTHER condition that must be met: as epsilon decreases to zero, delta HAS TO ALSO DECREASE TO ZERO. That’s where the failure occurs, and that is what would make these proofs more sound.

Your example of sin(x)/x is classic. In fact, most limits involving trigonometric functions are proven not with epsilon-delta, but by the Squeeze Theorem.

This has helped me a lot, thank you very much. :)

this helped so much! thank you!

ur the man …….

This makes sense.

You’re fabulous.

A + in Calculus for meee =)

love you too man

polygamy time! We all love you!

Awesome job on this.

YESSS. Thank you!

tht was an awesome explanaition!

wonderful! you real saved my dollars. your explanations are much clear, if it wasn’t for you i would have failed calculus.

Thanks, but please take time now to explain the delta neighborhood in 3D space and its relation to epsilon. Also, after selecting a neat working example, please choose one like

lim x/sqrt(x+y)

f(x,y)-> (1,1)

and show d-e proof. Need help. (did that problem come out aligned correctly? prob not)

Thanks

Thanks!!

THANK-YOU X 10 to the 6th power!!!! I have a week’s worth of precal and calculus homework to do before this friday so I can go to my cousin’s wedding and I don’t understand ANY of it!! It’s an online college course to boot…

Your explanation was so simple!!! I cannot believe it was so easy to understand!! Keep up the good work!! THANKYOUTHANKYOUTHANKYOUTHANKYOU!!!!!!!!!!!!!

This is the greatest and most clear explanation of this concept I have ever heard. I’ve been struggling with this for quite some time now I have it…finally! Thanks so much!!!

That is great! can you do that with the product property for limits? That written explanation makes all the difference and that’s what I need for the product property proof.

HOLY %$##%$%$$ YOU’RE MY GOD

I know you posted this a long time ago but it must be helping lots of people because it’s the 2nd listing on google (searching limits delta epsilon)! This is my THIRD calculus class and am just NOW understanding this bull! THANK YOU so much. If you are not a teacher, you should be because God has blessed you!

hello sir,this method is easy to understand and works really very good. thank you

I did not get it in class but I got it after reading the book. Im trust making sure that what I got is right and wow, you make it look so simple. Thanks so much, im bookmarking this blog and I’ll come back for others lessons

hopeful to see much more in future

it is so easy that i take this as a game where you take epsilons and get deltas. so it is all to work with epsilons and deltas and make fun out of it with the help of this method. thanks once again.

THANK YOU. You have no idea.

Thank you omg TAs and Calc Profs/Textbooks are useless…

Awesome blog

This is really fantastic. Four lectures of sitting through this nonsense explained absolutely nothing; and one 10-min reading did it. Now I’m able to do these problems and proofs. I still feel that the whole d-e concept is stupid and pointless, but this definitely explains it very well.

GENIUS. THANK YOU.

this is amazing. thank you so much

oh my god!!! i now understand ^^

how can we prove limit x->2 2x^2=8 using this method

kindly tell me how can we prove limit x->2 2x^2=8 using this?

awesome, that made so much more sense then what the professor dribbled out between yawns, thanks!

This was really very helpful.. U R a life savEr! (:

Thankyou so much, this is the clearest explanation of epsilon-delta I can find on the internet.

The IF-THEN logic of the delta-epsilon proof never really struck me, until now.

THANKYOU

I found this blog with a google search when I was trying to refresh my memory of how epsilon-delta proofs worked so that I could help a friend in their calc class. I used to tutor calc for a college, and these were the problems that made all of the tutors,myself included, run in terror everytime the calc 1 students would come in with these problems. Your fantastic explination really helped me remember what I learned 3 years ago and was probably better than the explination my professor gave then.

You have no idea how much I love you right now. No. Idea. Thanks so much.

I don’t understand why the professors couldn’t explain it like this..

thank you so much.

Zunduu ih tuslalaa daa ta bayarlala!! – You helped me a lot, thank you! in Mongolian

Saved me from another D or F on my calc test monday.

Hi guys. I have written an extensive explanation about the epsilon-delta definition here:

http://math4allages.wordpress.com/2010/03/15/epsilon-delta-definition-of-limits/

you may want to check it out. I would like to here from your reaction.

oh yes.problem clearly understood.please take us through the all calculus 1.thanks.

Hi Dave. I am looking for guest contributors and I really like this article.

I am just wondering, if are willing to be a guest contributor in my blog. If yes, i am just going to copy this and post it, then I am going to link it back to your blog and mention you as author.

How will you benefit?

This is one way of promoting your blog. I have an average of 10,000 viewers a month.

Very useful. Thank you.

However.

Seems to me though that unless you establish a relationship between epsilon and delta, then delta could be anything.

If this is the case then it wouldn’t prove much. Once they are related it means the closer a pt X is to a value c on the x axis, has implications for how near F(X) is to F(c). right?

Although what you have done is the easiest explanation so far that I have found, this area of math is another example of forcing students to learn by rote, and follow by example, something most will never truly understand.

I am still grappling with it myself. Maybe if there was some graphical illustration that would help.

This was very helpful. My books tell me what to do, but they don’t tell me how to think. You really managed to break it into little pieces and show not just the procedure, but how to actually understand the what’s and why’s. Much appreciated!

The meat of epsilon-delta proofs image does not show up on my mobile phone screen, all other images are fine.

Very good explanation. Thanks a lot!

Hi, I am unable to read on an iPhone the first and 2nd bubbles:

“before you even START with epsilon-delta . . .”

“The meat of epsilon-delta proofs is just this idea.”

The font colour in these two is very similar to the background colour. All others are fine.

Very great explanation. I understood it very quickly. It was extremely clear. Thank you so much!

YES!!! The plain old English bits helped a lot. Thanks!!!

You are a lifesaver!

Bobobobo – this makes it really clear. The link provided by Guillermo Bautista consolidated your post also.

Thanks!

Wow look at all the feedback you got; thanks for explaining this. The part about the inequalities helped me out a lot. I am forever grateful to you sir.

Your destiny has been chosen…become a Calculus Professor.

-God

Thank you, thank you, thank you, thank you, thank you, thank you. That’s all I can think to say. Wow. Very clear explanation. You did in 10 minutes what 2 teachers and 2 hours couldn’t do.

DOH, ITS SO SIMPLE AFTERALL

Oh, I love you for this. If I could somehow find out where you live and all of your personal info (in a non-creepy way, of course), I would camp in front of your house and sing songs of devotion involving epsilon-delta. Just in time for my midterm, thanks!

your a hero!

yo thanks man…..have a clac midterm tmrow…..u just saved my ass

Thanks a lot!

I’d marry you if you were a guy ahahah

Though I have to admit this didn’t solve every problem that I have right now, you really did establish the basic idea of an epsilon delta proof for me. Thank you sooooo much! Now if you could let me know how to do polynomials and trig functions and other stuff like that, that would be the greatest thing ever.

THANK YOU SO MUCH!!!!

Man, it had been said numerous times before, but having read through the link and parts of what you`ve written, I understood more in 10 minutes than I had during the entire course!!!!

THANK YOU :)

This is great! This is so helpful.

I have been trying to figure this problem out for months. A tutor and my teacher couldn’t explain this to me and I understood this in two seconds!

Can you explain a harder delta epsilon proof such as lim of x^3 as x-> 2 equals 8? My professor taught something about using a “constant c” that apparently we make up or something in order to prove it

This is… GLORIOUS!

Thank you so much. I don’t ever comment on stuff, but this deserved it.

I’ve honestly been trying to understand the formula for days until I came across this!

You explaining the absolute value thing finally made sense to me. I didn’t know what it meant until you explained it!

Thanks so much!

googling stuff rocks! but u rock more!

Thank you SO MUCH. I can’t even begin to tell you how much this article has helped me. I’m in calculus right now, but I’m horrible at math. So you can understand how this section totally tripped me up and made me behind in the class. Now that I’ve read this article, I can tackle my homework and hopefully do well on my exam coming up! Your article explained everything in such a clear yet precise way. Good work, and thanks again!

Thank you so much! You made it look so easy, that it’s easy to understand.

Thank you for this clear explanation of the epsilon-delta proofs. It certainly helps a lot of students struggling with this matter. However, the proofs are not completely correct. You do the proof in reverse.

You have to find a delta > 0 for every epsilon > 0, where epsilon can be made as small as possible. You have to proof that, for every epsilon-neighbourhood around L, you can find a delta-neighbourhood (one is already OK :-)) around c, not including c, so that abs(f(x)-L) < epsilon. You have to start with abs(x-c) < delta and manipulate it until you arrive at abs(f(x)-L) 3.

My trick is to start to choose a maximal value for delta, say 1. Nobody has objections against it :-) If delta_max=1, then abs(x+3)<7.

We know that abs(x-3) < delta (see definition delta)

Then,

abs(x-3)*abs(x+3) < 7 delta

abs((x-3)(x+3)) < 7 delta

abs(f(x)-L) < 7 delta = epsilon.

Choose delta = 1 or epsilon/7, but choose the smallest one.

In the proofs you give, if you choose a wrong value for the limit L, it can happen that there is nothing in your proof that says you that you're on the wrong track. In that case, you are proving sth that is untrue. However, in the proof above, it is impossible to complete the proof when you calculate the value of L incorrectly!

One of the previous comments said that delta has to go to 0 if epsilon goes to zero. However, that is NOT true, and that's why this requirement does not stand in any calculus textbook.

I hope this can be helpful. Kind regards. Steven.

wonderful, thank you!

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I don’t think the definition of a limit could be explained in a more clear way then this. THANK YOU. VERY MUCH.

You just taught me epsilon – delta in five minutes. Something my university professor has been trying to teach me through hours of lectures. What’s the point of university again…? Anyways, thank you very much!

can you teach my class? my teacher does not explain things like you do!

priyanka- thank u fr help

how can we prove that

lim 3x – 3 is not equal to 13 via epsilon-delta? could you explain?

x->5

So many thanks to you, like everyone else, I now got hang of this epsilon-delta proof mechanism.

<3

thank u so much..when ı afraid calculus and its exams,you become my rescuer ;) ı love u <3

Thank you! all my troubles gone!

I really understand it!! :D I will search now for other examples on your site. =) I like it!!

you are truly amazing

thenx for this

This is an excellent page, and I thank you for preparing it. However, why is there all the hostility toward this method of proof? The very last detail, the crucial bit which makes this a proof instead of a calculation problem, is that it demonstrates that there is a possible delta for every given epsilon (where epsilon > 0 and delta > 0). That is necessary for the limit to exist at all.

This type of example is chosen in calculus textbooks because the limit is easy for the student to see. Plugging in it is immediately clear that it is true, so you know you’re not wasting your time trying to decide if it IS true before proceeding. However, recall that in the definition of the limit that f(a) may not actually exist, yet the limit itself may. Plugging in will not demonstrate this at all and it will still be unclear whether there is a limit at all. The epsilon-delta proof provides the means to evaluate such a limit and be confident that it is true.

Yes, the example goes the long way about showing something that was immediately clear, but that’s what makes it an example, not stupid.

Thank you for this. Education should be more like this…

Please make it the example and equation for beigner(us)

i need extra problems and i invite you to solve

what about using this proof on rational functions? does the method still apply and would you use it in the same way or is there some other way to do it?

You are great to explain it in plain English. I have wasted so much time to learn it from books and prof notes.

please solve me lim x3-27/x-3=27 as x approaches 3

So δ is a measurement of the domain of a function where ε is a measurement of its corresponding range?

thanks ,its been a week and i have been trying to understand all this but yet you make it so simple.

This is incredible! I figure out what I haven’t been figured out in just a few minutes! Thanks a lot! Have you ever considered teaching as a career?

Your explanation was truly helpful. Thank you from the bottom of my heart!

Urgh !! i wasted bout 2hrs to understand dis crap on my own.. i wish i had read dis article of ur’s.. bt d problems dat i solved wer way more tougher dan the above..bt yea u made it look really easy. thumbs up !!

Very clear explanation.

Thanks, dude. You should tutor or something if you don’t already.

This was an amazing explanation. Hmph, it was simpler than I thought. Thanks!

You are amazing!!!! Thanks, thanks, thaaaaaaanks!!!!!!!!!!

HONESTLY SO HELPFUL. THANK YOU!

i love u man!

There’s nothing wrong with this proof. The (incorrect) situation where L=13 yields d=(1-e)/3, as stated above. However, the condition that prevents this from being correct is given in the lesson. “The limit above exists if and only if for each ε > 0, there exists a δ > 0…”, this is not satisfied, as for any e>1, d<0.

Finally understood this eplison delta bs! Thank you for your time in this, it really helped! Now I can finally go to sleep

Thank you so much!! Will you re-write my entire math book? :-)

umm i got little bit tho well done

Excellent!

Youre the fucking man dude

you are god.

i have been a failure all through in this part of delta and epsilon now i understand the basics from your explanation thank you

Such a win. Make more.

Awesum explanantion..

this was really helpful………..i have a test tommorrow on this stuff and limits in general…..could u explain to me the squeeze theorem? but your delta epsilon explanation was so very very helpful. thanks so much!!!!!

Putting it in plain English (the “|x-a| x is within delta units of a” bit) not only helped me understand more concisely the concept of delta-epsilon proofs and how to show them, it also helped me remember that it’s |x-a|delta because I’ve always had trouble remembering whether it is greater than or less than! You’re the best! Thank you so much! I have my calculus midterm for first year electrical engineering tomorrow and I thought I was screwed for this part of the exam but now I feel more confident in this! You’re awesome!!! :D

Something is wrong with the comment thing…it took out all the greater than/less than symbols and the arrow…well hopefully you’ll understand what I meant lol anyway yeah thanks again!!!

there are two meanings of “understanding” epsilon-delta.

1.to comprehend the meaning of epsilon-delta.

2.to know why epsilon-delta is formulated at the first place.(Why it is needed to be precise).

Students are more interested in taking 1 rather than 2 which is important to appreciate why mathematics is needed to be rigorous.1 are for those who want to pass the exam while 2 are for those who posses the hunger for knowledge.

Too bad students are more practical than of a curious type which is suppose to be.

Good

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