epsilon-delta definition ultimate introduction

My ultimate introduction to the epsilon-delta definition of limits in calculus! The epsilon-delta definition of a limit is commonly considered the hardest topic in Calculus 1 (it's also the important part at the beginning of real analysis). The best way to understand this precise definition of a limit is to actually use an actual value for epsilon first, then solve for the corresponding delta. When writing epsilon-delta proofs for limits, be sure to set it up: given, choose, suppose, and check!

Check out more examples of proving limits with the εδ definition:

0:00 the εδ definition of a limit is the hardest topic in calculus 1
1:34 limit of sqrt(2x+1) as x approaches 4
4:14 if epsilon is 0.2, how to find the biggest delta
10:27 writing the εδ proof

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[blackpenredpen]

Check out 24 more rigorous proofs: ultimate calculus: 24 rigorous limit proofs

[redblasphemy]

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

[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.

[JohnSmith-rftx]

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.

[coreymonsta]

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.

[sadiakhan]

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!

[outofnowhereboy]

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!

[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.

[dellta]

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!

[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.

[marcelandrade]

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

[greghansen]

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.

[uvxv_]

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

[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.

[scottwitoff]

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

[alexj.]

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

[abhishekkumar]

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

[kaylo]

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.

[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).

[MiracleMirror]

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