How to do epsilon-delta proofs (ultimate calculus guide)

[@blackpenredpen]

Some minor mistakes: at 1:31:41, please change all the ε to M at 1:42:28, it should be 1/abs(x-3) is greater than 1/δ. And then say 1/(abs(x-3))^2=1/(x-3)^2 is greater than 1/δ^2 at 1:48:26, we need to add the condition that N is greater than 1. So please write N=max{1, 2M} Thanks to everyone who pointed these out. Here's my hand-written notes for all these 24 proofs: (Patreons only) www.patreon.com/posts/notes-for-24-88670205

[@ochukoobrikogho5587]

Very Good job

[@karimalramlawi7228]

Another mistake. You need to write N=max{ 1, 2M} Not max{1 , 2epsilone}

[@Will-Ch]

:v

[@bartekabuz855]

I'm missing some harder limits involving sin(x), e^x, log(x) etc. My class never covered that. Part 2 with those please

[@mixedgrene9729]

My class also never covered exponential/logarithmic limits. It may be that they are not part of calc 1 because they're outside the scope of the class. It would be cool to see how that stuff works though

[@LordBrainz]

Trust me, they're too hard. Use sandwich theorem for them, it's more pleasant

[@bartekabuz855]

sorry I'm not a pussy to say something basic is "too hard"

[@rodrigoaleman1072]

This video, like the "100 limits" and "100 derivates" are good asf, you're a savior teacher!!!!!! Thanks a lot

[@ngchorhong3174]

Hi again bprp, I was the STPM student from before in your 100 derivatives video. I just want to mention how much you have helped me as I am about to tackle advanced calculus next semester. I am sure many others are grateful for these videos as well. Really appreciate this entire step by step series. A lot of students want this sort of teaching, holding their hands through every small detail but it is just not practical in class as it takes up too much time from both ends (also as a student I am ashamed to ask questions). Therefore, putting up videos for us to follow you "live and in the moment" really benefits us and is very time-flexible. I have been pausing before each solution or outright try to write faster to familiarize myself with the steps and it has worked wonders for me. The fact that you took the time to write out each and every line for long periods of time still amazes me (of course these 2 hours must have been just a warm up for you by now, considering other 100---something videos). Thank you once again for these passionate and helpful videos, stating out the "do's" and "dont's" as well as the "what-ifs", all whilst making math understandable for us in an entertaining and calm way. You can expect me in other videos :D Here are some moments 9:23, I can imagine two kids(inequalities) "playing" tug of war with the limit being pulled 11:25, I understand that box brings a lot of satisfaction , same thing when you close all tabs of whatever you are searching after an assignment is done. 1:07:55, two kids are back 1:17:47 , your face :D , my face -_- 1:18:27, "If you do it like this then N will be the best end for this proof" , this is shown in the subtitles Best end as in a sarastic way to say "just leave it for somebody else to do" / " you screwed up" / "Drop your pen" 1:24:03, me when cancelling terms just feels too good 1:26:16, imagine if it stopped recording (suggestion to check every 30 minutes or so...) 1:38:15, fear-->relief in 1 second, proceeds to laugh while verifying. Highly relatable in exams especially overchecking with the calculator for two digit additions XD 1:40:55 ---> 1:41:12, (raises hand) Thank you so much, your attention to detail and awareness of perspectives from students is one of your admirable qualities. Keep it up! 1:44:10, (scared voice) Why?, what about negative infinity... Uneasiness follows when you smirk like that 1:45:18, ah s*** here we go again 1:51:22, Good ! Now do 100 🙂 Finally we can rest now. See you next time

[@ilovecats0998]

This made me so happy to read! I'm just starting out calculus and am understanding proofs in depth because of his channel! Wish you well 😊

[@maruferieldelcarmen9573]

Really helped; I used to think the rigorous limit definition was really complicated. I'm getting ready with entering graduate-level maths. My undergrad was in engineering, so a bit shaky and my graduate coordinator said there's gonna be a big paradigm shift in thinking, but this is already a good start.

[@kornelviktor6985]

Lets goooooo. The school just started so I can flex with this knowledge. 😂😂❤❤

[@CKNGAI-r8x]

Be cautious people may start calling u a nerd

[@kornelviktor6985]

@@CKNGAI-r8x I'm already called a nerd sometimes so I don't really care

[@46-suvendusamantaray49]

Here in India , if you are a topper , everyone respects you.

[@WritersDigest-b8f]

It is easier to understand difficult calculus concepts by watching examples being solved because u actually see the established rules, theorems being applied to a situation to solve the problem, and then it becomes easier to understand & hard to forget

[@JonnyMath]

Thanks professor!!! Fortunately now since it's September as maths channels we'll get more views!!! Thanks a lot I love your videos!!!🥳🎊

[@sethlupo4736]

Thank you 👍👍👍. I took an analysis class a year ago and we did a bunch of these but I realized that I was out of practice... but going through this video gave me all the practice I needed!

[@joekerr3638]

Thank you for doing this channel. You make calculus fun and enjoyable.

[@Emanon0707]

Thank you very much! I have a test this Friday and this video really helps me a lot !!!

[@nbayoungboydagoat5376]

Bro you are the goat for saving me I was struggling with epilson delta and now I have finally grapsed the concept

[@rada4917]

i finally understand delta epsilon proofs. ur amazing.

[@ulisses_nicolau_barros]

Thanks, bprp. Those proofs are really important in Calculus, mainly for deeper students or math (Uni) students. Gonna revise from time to time.

[@federicopagano6590]

1:35:15 here its another way to avoid guessing . First off ,you find taylor series of I F-L I around x=-2 .then you apply I A + B+CI Is less than IAI + IBI + ICI where A,B,C are powers of I(x+2)I meaning this sum is bounded by powers of δ. then because 0 < δ < 1 , δ^n < δ. then factor out δ and you end up ALWAYS with δ(SUM Abs ( taylor coefficients) ) < ε. you end up with δ=ε/4.This not only applies to polynomials but to rational functions where we can use geometric series to apply the same trick

[@nonentity168]

Wonderful lecture. May you have good health for all the best.

[@Emmanuel-rr3fz]

youve been nothing short of an inspiration to me i used to follow you on my previouc acc and youve inspired me to keep going thank you

[@Andyg2g]

This video was so so so helpful. I only wish it existed back in 2015 when I was taking Real Analysis.

[@MarkPaul1316]

I really liked the video. I learned a lot on your channel. Continue with your wonderful work.

[@Mihrettsinfie]

I was struggling epsilon delta finally got it thanks

[@WritersDigest-b8f]

This is a different approach to prove limits First u ensured M exists in terms of δ & the expression > M in terms of δ (Τhe expression is ensured to be greater than M in terms of δ or N or < ε in terms of N or δ) Then δ is pulled out of it from RHS of the expression and then δ in terms of ε plugged in to ensure the greater than or less than inequality.

[@sarabicaku2205]

I have an exam in a week 😭...you uploaded it at the perfect timing

[@siddharthapahari5774]

watching still at 3:26 am (preparation for calc 1 midterm) thanks a lot for this video!

[@oqardZ]

From 1:31:41 to 1:32:10 (problem 20) replace epsilon with M everywhere.

[@complexlogarithm3993]

Thank you, I was wondering about that

[@blackpenredpen]

Yes. I mixed that up. Thanks!

[@complexlogarithm3993]

@@blackpenredpen Btw, I kept watching. Thank you, it was a wonderful practice for my real analysis course.

[@jasmineokoror42]

I am so grateful for what you do

[@oncanvas1977]

Hi. Thank you for making this video. It helped me to understand the epsilon-delta proofs. Can you please make a squeeze theorem ultimate guide?

[@Jad141]

For the last question, choosing N=2M allows for 01). I love your videos, keep up the great work❤

[@blackpenredpen]

Ah thanks!! Let’s do N=max{1, 2M} so that >1 is more clear

[@steamhunter7504]

​@@blackpenredpenWhy do we choose max instead of min? Nice video!

[@tegakomori5401]

This was so effective Thank you so much

[@Victual88]

Thanks a ton my friend, watched the whole way through😃.

[@dysphoricjoy]

You know you're a nerd when this video drops and you audibly say "hell yeah!"

[@jaywardhanraghu4822]

THANK YOU SOOO MUCH I WATCHED ALL THE WAY THROUGH 😍😍😍😍❤❤❤❤

[@Γιώργος-ε6τ]

Very nice video, was worth watching till end. Like other people I too hope you make a similar one in future with trig finctions and exponentials

[@JB-ym4up]

#23. The direction that x approaches 3 from is not defined so when you assume abs(x-3) is positive and take it outside the abs you are only proving the x approaches 3 from above limit. The fact that in the original equation it is (x-3)² is not relevant to the absolute value situation. You must also prove the x approaches 3 from below case.

[@philip_fa]

1:41:09 I'm studying computer science and for my studies and I have to take analysis 1 course. So yeah I've been watching this video until 1:41:09 and I'm leaving a comment to let you know that I've been practicing with your video for the last few hours :-)

[@sadi_supercell2132]

On question 15 you could solve it too by factoring x^2 - 1 so it would be x/(x+1)(x-1)

[@SHErlock_plusplus]

SUPERBBBBBBBBBB!!! what books do you recommend for more explanation and exercises on real analysis? Thank you!!

[@charlydramais5020]

you just saved my year

[@kuachkoangchol8961]

you are very intelligent person

[@WritersDigest-b8f]

U advised me earlier to go over this YT. I went overit thrice, and clearly understood everything, thank u. Could u suggest which should be the next calculus YT from u that I should go over? 100 calculus 1 problems or 100 calculus 2 problems or 100 calculus 3 problems. Or something else?

[@sohampinemath1086]

Day 1: 100 Differential equations please. You are the greatest teacher

[@gabrieldalmolin866]

In the last exercise, can't you get rid of the 1 of the denominator? You could say x2/(x+1) < x2/x, simplify the x, you compare x with N, x is larger than N which is true, set N to be equal to M? I'm just wondering

[@VaibhavSharma-zj4gk]

Hello professor, please make 1 video for examples in which delta epsilon defination fails because the limit donot exist there.

[@blackpenredpen]

Do you have a specific limit in mind?

[@VaibhavSharma-zj4gk]

@@blackpenredpen limit of sin(1/x) at 0.

[@WritersDigest-b8f]

Could u explain the following? Prove Lim x tends to infinity 4x2-3x+2/8x2-6x+1 =1/2 Tried to understand it with the help of someone else, it’s difficult to clearly understand. Thank you

[@mr.inhuman7932]

Thanks for your work.

[@WritersDigest-b8f]

Please suggest a Calculus book that explains all calculus concepts with examples and solutions. I learned δε, ΝM, Nε, δΜ limit proofs by looking at so many examples u presented on YT, thank u. I want to learn FFT concepts with many solved examples, so far I have not found this opportunity on YT. I am hoping u would teach FFT on YT. Are u going to teach FFT? Please do so. Thank u

[@WritersDigest-b8f]

I got the 4th edition of calculus by Michael Spivak, but not sure how to make the best use of it before the return due date. Any suggestions? Should I try to find the solution manual? Seems, my best option is to watch KZbin videos on calculus.

[@JGambrosia]

11:05. Im kind of confused with the less than or equal. What if we choose epsilon to equal to 0.2? Meaning that 2*epsilon = 0.2 so min(1,2*epsilon) = 1, where 1 > 0.2 so 1 > 2*epsilon, making min(1,2*epsilon) > 2*epsilon. Am I wrong?

[@harshtalk007]

Please teach integration by graph and do also teach all graph like hyperbola graph of mode etc

[@jeffreychu2940]

Great vid. But for q15, why not just do x/(x^2-1) < (x+1)/(x^2-1) = 1/(x-1) and continue the proof from there?

[@MajesticQT]

in the 51:50 proof, you said we can really choose any value other than 1 for the min{} but wouldn't that change what delta is? In my attempt I choose delta = min{ 2, epsilon/2}. is this still valid? Or does the number I choose have to be less than 1?

[@karimalramlawi7228]

No it's not valid because here we say that x can't be 2 so if the delta is 2 then x can be between 1 and 5 but that contradicts our statement because 2 is between 1 and 5. In that case you need to choose a smaller delta.

[@johnnydepp4866]

Hi bprp, I wonder if this course will also cover Cauchy sequences and continuity, especially with multivariable functions?

[@justinpark939]

1:13:16 you could also add a plus 1 to the numerator and cancel the factors :) Here is a cool limit I found from generalizing from a problem in my text book Show that the result ln(cos(ax))/ln(cos(bx)) as x->0 is a^2/b^2 (you may use the fact that lim x->0 ln(1+x)/x tends to 1) Bonus question:do we need a epsilon delta proof to confirm with more rigor?

[@teddygaming4076]

use L'H twice, but use multiplication law in 2nd one as well

[@justinpark939]

@teddygaming4076 fair enough I guess without lh would be what I was intending.

[@thexoxob9448]

Let's try limx->0 sin x = 0. Given ε > 0. Choose δ = ? -> δ=sin⁻¹ε Suppose 0 < abs(x) < δ Check abs(sin x) = sin(abs(x)) for around x = 0. < sin δ since sin is inc. around x = 0. =sin(sin⁻¹ε) =ε. ■

[@Maraq369]

Q19: Can N = 1/ ε Instead of N = 1/sqrt(ε) ?

[@blackpenredpen]

Yes. That’s fine too.

[@ИринаГригорьева-ч3я]

Are \infty and +\infty the same? I think no. So in #3 we have to suppose that |x| > N, aren't we?

[@briskioO]

god im STOKED for this

[@Mephisto707]

53:39 N has to be bigger than 1, not zero, for x2 to be bigger than N2.

[@blackpenredpen]

I think since x^2 is an increasing function when x>=0, so N>0 should work here. But maybe there's something I am not seeing, please correct me if I am wrong. Thanks.

[@Mephisto707]

​@@blackpenredpenYou are right, forget what I said.

[@abemt4088]

respect brother

[@blackpenredpen]

Thanks.

[@nikoletaiakovatou6853]

in example 13 why not take |(x+2)|^2< 25 and take just for x?

[@WritersDigest-b8f]

Pb 20 Should it not have been >M Could u please look at it again Thank u

[@varunraj8606]

thank you so much sir

[@Maraq369]

Q20: Did you mean to use M instead of epsilon? Given M>0 and then delta is equal to 1/3ε Is delta supposed to be equal to 1/3M?

[@metalsonic8859]

M = eps He used M at the start because in definition of limit that goes to inf, eps is supposed to be infenetly big (not infinetly small as ussual), so people prefer to use another symbol in this definiton (like M there). He just forgot to further use M instead of eps, I suppose

[@Mathematics-with-MohsinRaja]

Its mean there is no option to mark attendence it will be automatically counted when you complete video...and for some subjects there is no attendance

[@manstuckinabox3679]

1:02:27: I was wondering if from the fact |x-2|

[@JirivandenAssem]

59:35 could've just rewrite it as (x+1)^2 +3 and then showing x+1 is less than 4, is a bit faster.

[@mrvzhao]

It's funny how youtube pushes shady advertisements for "quantum stock trading" programs along with your videos 😀 I think youtube needs to give your viewers more credit than that...

[@sherkki9006]

I am still watchin!

[@blackpenredpen]

Awesome!

[@WritersDigest-b8f]

Could u solve this one 4x2-3x+2/8x2-6x+1 =1/2 N & ε case Thank u

[@multilingualprogrammer3154]

You should find the derivative of x!

[@tomtomspa]

number 15 wich was a little cumbersome it would've been way easier if you had replaced at the numerator x with x+1 wich is bigger. Now you could've simplified the factorized fraction to get an easy 1/(x-1)

[@tomtomspa]

the same for number 24

[@ZivAsh-t5b]

thank you!

[@gdmathguy]

what about proofs for limits that don't exist?

[@ethemesonmez]

thank you. after this I decided to take analysis lol

[@stillpresident2384]

1:23:51

[@WritersDigest-b8f]

Pb 24 Why is x>1

[@iambezapp711]

you are the greatest 🥰🥰🥰🥰🥰🥰

[@BusyBlueLion]

You are fantastic

[@kriswillems5661]

Why not just take the slope at that point and take delta

[@michaels333]

The derivative is based on the limit existing at that point, and this is what you’re trying to show (rigorously).

[@gavinphillips392]

WE LIT!!!!

[@fasebingterfe6354]

12:06 Don't mind me, I'm just marking where I left off

[@willie333b]

Why d owe need to put min as 1

[@adamdaouk8673]

#24 I wrote x²/(x+1) > (x²-1)/(x+1) then simplified and got >x-1>N-1>M for N=max{1;M+1} Is it correct?

[@PeterCrasher1]

yes got it too

[@DivyWadhwani-k3p]

I concur

[@antipsikiyatriKizi]

Why do we have the minimum for delta as 1?

[@abdullohabdurrohmanm1003]

hi bprp try to give an example of a limit that can't be proven

[@Will-Ch]

genial bro , mas videos asi porfavor de todo el cálculo. thanks

[@iiwacky6480]

Can you do a proof of a limit that doesnt exist please?

[@엘라-r8k]

Thanks

[@Poppinwheeeeellllllieeeeez]

Do you know how to rebuild a motorcycle engine? Restore a vintage audio amplifier? Oil painting or play the bass? Downhill ski?

[@bernarddoherty4014]

Lim bprp->24 =🏆🏅🏅🤗

[@di-riso]

On the last question couldnt you just drop the x2

[@attiliolesilio55]

(1) 2x-3 is continous function and then lim=function. Il resto è fatica sprecata 😂

[@KentaEvans]

Yes!

[@di-riso]

At q 15 i go √1/epsilon +1

[@salvadorsabateromero2258]

55:12

[@ew9072]

how did we correlate delta to M at 21:35 😭

[@salvadorsabateromero2258]

11:41

[@josephmargaryan]

what level math is this

[@JGambrosia]

High school AP calculus or first year university calculus 1.

[@thatomofolo452]

Limits are cool ✌️✌️😎

[@strikerclub1233]

Hi

[@darcash1738]

Maybe it’s just me, but I understand how the questions work but have no clue how it’s proving anything 😂