epsilon-delta proof ultimate study guide

[@bprpcalculusbasics]

New video: Your first and second day in college calculus: kzbin.info/www/bejne/fXvXfn6fqqaUe5Isi=GTBLG_F3moYnhvJq

[@annaholm9704]

Oh my God😃 This guy is a godsend. I'm so glad he made this video! I'm only in calc 1, and because of him, I already know the lectures before they are taught.

[@kingbeauregard]

Can I say a conceptual thing? Imagine you've got your function f(x), and there is a point (a, L) on the function and you want to prove that it's continuous. So, imagine a rectangle centered on (a, L), and it is tall enough that the function never touches the top or bottom edges of the rectangle. Now, can you resize the rectangle down to nothing, without the function touching the top or bottom edge at any resizing? That's what epsilon-delta is trying to do, abstractly rather than visually: the rectangle would be 2*delta wide and 2*epsilon high. So you're just working out the proportions of the rectangles; if you can derive proportions for the rectangles, the function is continuous.

[@iteoluwaoladejo4240]

@@kingbeauregard W concept. I never thought of it that way. Thanks a lot for sharing.

[@josefmakili4062]

To dopa nena

[@IncendiaHL]

My mathematical analysis exam coming up, I realized how rusty and inexperienced I am with these. There are plenty of videos explaining the concepts, but that's not the hard part. The hard part is actually getting experience in using them, and this video!!! Is!! Exactly!!! What I wanted!! An incredible teacher just trucking through a boatload of Proofs. Thank you so much!

[@batteryjuicy4231]

this man is an absolute legend. When he learned this stuff, there was no black pen red pen and when he learned it so well to create this channel to teach us bozzos

[@leseagleexe6947]

Youre an absolute gem sir

[@bprpcalculusbasics]

Thanks!!

[@omerdogancanbilgili7401]

bro you literally save my academic career thank you

[@wesc6452]

This is very helpful! I am in intro to proofs this semester but also in an analysis class that is graduate level so it has been challenging so far. This really makes things much more understandable - thank you!

[@hyde_stopStealingMyUsername]

Thank you so much, at first I didn't understand anything but because of your explanations I was able to solve the last 3 by myself

[@antipsikiyatriKizi]

Best video about this topic on KZbin.

[@nvapisces7011]

It would be great if you could make a video on a guide to prove one sided limits as well as infinite limits and a mix of both. Then make separate videos proving the different limit laws (sum, product, quotient, power, root law)

[@rain-uw9ju]

isn't epsilon delta the method used to prove all of the different limit laws? I thought it was the basis for limits in general.

[@alibinnaseer]

thank you soo much!!! I watched sal's video, then yours and then again sal's video and then went to ChatGPT and then came back to this, rewatched it and now i finally understood it !!! May Allah bless you !! 💖💖

[@eddiephysics]

Condensed and extremely easy to follow along!

[@luciekiss1618]

Thank you for your passion ! This video is so helpful. Keep doing what youre doing.

[@user-qj3rv2mo1b]

At the end of each proof u prove it by deducing δ to get ε. It definitely makes sense, And grateful to you for wonderful explanation and so many interesting examples. Now I feel comfortable with proving limits using your GCSC process. However, I have also seen few other proofs, they determine δ like u do and then back track through the key controllable factors to the start, to show |F(x)-L|

[@danni4451]

God, I learned a lot from this video. Thank you so much! You teached so well. I couldn't even understand how this actually works, and after watching your video it makes me 恍然大悟.😁😁😁😁😁😁

[@bprpcalculusbasics]

Glad to help!! 😃

[@user-oh4tf8lj9y]

you are amaizng toutr i hope you are my proffessor instead of my profferssor THANK YOU A LOT

[@Sdw_Bld]

How is this guy such a good teacher?!

[@del3t3d]

Wonderful, exam tomorrow and you simplify things so that they can be easily well understood :)

[@bhavishaysharma8648]

Okay. This was crazy!!! If you read this, thanks a lot bud :)

[@bprpcalculusbasics]

Thank you!

[@kingbeauregard]

I notice a pretty common practical methodology on each of these, almost a road map for what to do with the math: 1) You start with "f(x) - f(a)"; find some way to mingle those two terms, such that an "x-a" emerges in there somewhere. 2) Separate the "x-a" from the rest of the expression, so you have something like "(x-a)*(other stuff)". That "x-a" will become our delta. 3) Get rid of any instances of "x" that appear in "other stuff".

[@user-vr2ny2ff9u]

You are outstanding, thanks you, i understood this !! In my institution no one explained this...

[@Akane_Kana]

This video explained everything. Love that!

[@macdaveshub]

I love your teaching sir 🥰🥰 This is a topic I have been finding soo difficult to understand. But you have elaborated it completely. And I have learnt it. Keep up the good work ✌️🥰

[@emmecuh]

Great video as always, would love to see some one sided and infinite limit proofs someday.

[@leonardobarrera2816]

Thanks If you don’t want to do the problem that I am solving for x^(x+1)=10 Don’t do it, I will se how I can solve it But thanks about all your videos Loves and prayers from Ecuador

[@bprpcalculusbasics]

I don't think that's doable algebraically. We can use Newton's method, or graphical method. Here's the answer on WFA www.wolframalpha.com/input?i=solve+x%5E%28x%2B1%29%3D10

[@leonardobarrera2816]

@@bprpcalculusbasics Thanks a lot for respond my comment, I don't know, but I would like to go to the event that will be in North California, because I love maths, and also you will be there, and it will be great for all your subcribers that also will be in there, and I would like to subscribe for the event. Thanks a lot for the feedback Remember, you are genious, I would like to be like you, I guess, if the things go in the right way, I will be like you!!! Cheers!! Loves and players from Ecuador!!!!!!!

[@SuperYoonHo]

@@leonardobarrera2816 u soUnd like sir premath LOL

[@leonardobarrera2816]

@@SuperYoonHo really???? I express like that because I didn’t now how to say like we say in Spanish!! Thanks, I would like to be in the tour mente of north California (Steve send in the channel of blackpenredpen)

[@SuperYoonHo]

@@leonardobarrera2816 Oooh that's cool! 👍👍

[@mecidgaliba2854]

Thanks a lot for this comprehensive video. I was just wondering if you could make a video about how to write down this proof in exams. I mean not by using arrows or red markers. Thanks in advance!

[@ngozikanwachukwu6623]

honestly, I really don't like the epsilon delta proof at all. But thank you for making it easier and tolerable for me❣

[@yakobbeyene8764]

Thank you for your help! Would you proof this question by using epsilon and delta definition. Limit x approaches 1 ( 5x^5+4x^3+x^2+x+1=12

[@SuperYoonHo]

Thank you so much!

[@ronycb7168]

21:00 we could just get rid of only the square root like you had shown in another video, I just want to see if this is also correct

[@ryanfox6260]

I've been exploring a different technique perhaps you can comment on. I would set up the equation 4+e = 6(x+d)+2 and solve for d = e/6. I visualize it as setting one of the bounds on the y-axis equal to the outer bound on the x-axis. In general it's L + e = f(x+d), where you know values of L and x. thoughts?

[@stevenstone7101]

Can you explain further?

[@Music-zx4po]

Thank you for your work! I could follow your first example, however why exactly may one introduce the epsilon - what exactly permits this step? This is probably like proofing the premises.

[@mikesmithyes967]

Do Measure Theory practice problems.

[@markobavdek9450]

Torture of mind with these letters. Anyway I'll watch this...

[@Yt-ff6hn]

Right said ...

[@jeloabmelak9783]

Really helped me save the semister🎉❤

[@cyrusdeng0108]

keywords： Given, Choose, Suppose, Check !

[@monsieurlambertz7706]

I'm sort of confused. When he says a/b0, it's all fine. But I've been thinking what if a=b? If a=b then a/b = 1 < a, so in this case a must be greater than 1. In other words, can I just assume a/b < a in all cases? Even when |x-a| = sqrt(x) + sqrt(a)?

[@sihlec1047]

24:19 I'am dead... use pi... use 1/e

[@kingbeauregard]

What if we were doing f(x) = e^x? Here's how I think it goes, but I'll show you where I run into trouble: |e^x - e^a| < epsilon e^a*|e^(x-a) - 1| < epsilon |e^(x-a) -1| < epsilon/e^a -epsilon/e^a < e^(x-a) - 1 < epsilon/e^a 1 - epsilon/e^a < e^(x-a) < 1 + epsilon/e^a Next step is to take the ln, which I can do because ln(x) is an increasing function just as e(x) is increasing. In other words, it's not going to flip the sign. BUT -- you can't take the ln of a negative, which means that epsilon must be less than e^a. Right? This is ultimately my question, but I'll continue the arithmetic. ln (1 - epsilon/e^a) < x - a < ln(1 + epsilon/e^a) Now we want to put this back in absolute value form: |x-a| < ln(1 + epsilon/e^a) ... and I took the "plus" term rather than the "minus" term because its magnitude is smaller. Finally, we have: delta = ln(1 + epsilon/e^a) In broad strokes, this holds up: since epsilon is always positive, the ln() is always positive; and when epsilon equals 0, delta also equals 0. But again, the epsilon-delta definition specifies "for ANY epsilon", and I'm only allowing 0 < epsilon < e^a. So, is this an invalid proof, because I'm specifying a maximum epsilon?

[@dalek1099]

If mod(e^x-e^a)< epsilon then it is also less than epsilon' where epsilon'> epsilon so the maximum epsilon isn't a problem.

[@magnetic_monopole]

This helps me a lot!

[@kiri0725]

Thank you very much !!!

[@heroasik5423]

in limsqrt(x),x----->4 delta=4epsilon

[@user-qy9py4lv9e]

Loved itt!

[@ishanschneider9289]

absolute legend

[@Firefly256]

What should I do if I can’t get the |x-a|? For example, [limit of 2x+1 as x->1] = 3 0 < |x - 1| < delta |2x - 1 - 3| = 2|x - 2| |x - 1| < 1 -1 < x - 1 < 1 -2 < x - 2 < 0 |x - 2| < 2 So I can only get to: 2|x - 2| < 4 I don’t have delta in the above, how should I approach this?

[@fretzyjanebares2257]

Hello, I think you have incorrectly written I f(x) - L | in your proof. You wrote |2x - 1 - 3| instead of |2x + 1 - 3|. The 3rd line should be |2x + 1- 3| = | 2x -2 | = 2 | x-1 | < 2 * delta. Hence delta is epsilon/2. Hope this helps. 😀

[@Firefly256]

@@fretzyjanebares2257 oh right 😅, so does this situation never appear at all for other functions?

[@dearlymats428]

The most humorous lecturer in my life 😅😅😅😅😅😅wow I enjoyed the lecture , best lecture in my life ,you make it easier to understand with jokes # if you want to be adventurous put something like pi or root of 17 got me offf😅😅😅😅😅etc

[@waldomonroy452]

Right on time

[@Drum11craze]

Can you do a video showing a proof of a limit as x goes to infinity? Can't find anything on this subject.

[@TazzD83]

lucky that I found you!! 😁Thanks a bunch!

[@NWS390]

It's grate sir❤

[@stewartgarikayi4560]

Thank you Professor

[@jtlbb2]

Something doesn’t make sense. Around the 40 minute mark you had (x-2/2x) and you split it up into (x-2)(1/2x). Previously in similar problems you said to ignore the denominator because the numerator was greater than the whole fraction. Why is this different? After all (x-2/2x)< (x-2), so why is this different than problems you did earlier?

[@bprpcalculusbasics]

Could u please give the exact time stamps?

[@jtlbb2]

Of course. At 16:00 and then 39:09. I’m confused why the approaches are different.

[@jtlbb2]

@@bprpcalculusbasics I think I might’ve figured out why. Is it because in the problem at 39:09, you can’t be sure the denominator is greater than 1? For example, the bottom could be 2x = 2(0.25) making it less than one meaning the fraction is larger than just the numerator. I think I just figured out the difference and didn’t think it through enough before asking.

[@bprpcalculusbasics]

@@jtlbb2 Yes, exactly! If it was a 2+x on the bottom at 39:09, then we could also ignore the denominator. Also, if you would like to see other examples with x->inf or the limit being inf, this video is more complete: kzbin.info/www/bejne/d5fVn4yJarhrjKcsi=V8rMFSOiXrsenyz6

[@jtlbb2]

@@bprpcalculusbasics thank you. I subscribed because you’re a very good teacher

[@kaistrandskov]

This is a little confusing for me. I prefer to do this: 1. We need to figure out how to choose a delta, when given an epsilon. So start with |F(x)-L| < epsilon, and plug in all the numbers for F(x) and L. This is what we know because we are given an epsilon. 2. Next, figure out how to choose a delta based on epsilon. Now, get this inequality into the form of |x-a| < G(epsilon), where G is some junk involving epsilon that we just solved for. Since our goal is to choose a delta so that |x-a| < delta, we can just choose a delta = G(epsilon). 3. Now go through the proof from the beginning, and choose a delta = the G(epsilon) we found earlier.

[@tarundeeparora865]

love you

[@alpasamudra3479]

sir please do calculus 3 multvariable case

[@evanssmith1327]

Man please do you have moment of forces channel??? You are just good

[@alexei2985]

At 45:50 you write down 1.7 as if it would technically work, then you suggest to use 1 instead because it is easier. I’m confused why 1.7 would technically work because when I use 1.7, I get a different answer at the end. Why would 1.7 work if it gives us a different answer?

[@giana.2381]

The epsilon-delta definition of a limit only requires us to prove that a delta (such that the conditions are satisfied) exists. It doesn't require us to find all delta that satisfies the conditions. It also doesn't require us to prove that the delta is unique. So having a different delta is fine, as long as the conditions are satisfied. Sorry, this might not be useful to you anymore since I'm replying to you a couple of months late. 😅

[@creepsow1236]

I love the shirt

[@cunningham.s_law]

thanks can you do one where you prove inf and sup of a bunch of sets

[@energy-tunes]

by the supremum axiom all finite subsets of real numbers have a supremum. then you can easily prove that they also have infimum given this axiom

[@tahastm2522]

hello thanks for your video can you help me to solve lim(x^1/2+1)x=1 =1?

[@heroasik5423]

graphically study the epsilon delta you will get simple epsilon/delta=f'(a)

[@niteshgiri8369]

lim x -> - 2 (x ^ 2 + 3x) = - 2 solve it by delta epsilon definition

[@KIYAFey]

what if as x goes to infinitive instead of constant number

[@user-bd8gv6po5w]

Appreciate that a lot sir

[@Paparazzi1e]

This is cool 😎

[@MicaelSLopes]

Nice algebra class

[@Professor-Q]

I love you

[@velmagiah4310]

Amazing

[@paulpitters2185]

You’re so smart

[@thomasjefferson6225]

Helped me in analysis a lot.

[@salvadorsabateromero9707]

21:40

[@salvadorsabateromero9707]

13:48

[@retro_cody_]

This is really cool. So it’s basically mathematical manipulation 🤣. Finishing up calc1. I somewhat get the proofs but ofc need some practice and help :)

[@energy-tunes]

were you recording this for hours you sound like youre going to pass out in the entire second part

[@salvadorsabateromero9707]

32:43

[@user-kh6cm1kd6y]

14:40

[@DBD253]

Man this shit annoyed the hell of me. Wish this video was out during covid when i was finishing my degree advanced Calc kicked my ass a bit...

[@heroasik5423]

simple epsilon/delta=f'(a)

[@rafiqqayumi5957]

you are wrong in first suppose part

[@salvadorsabateromero9707]

31:27