How to do epsilon-delta proofs (ultimate calculus guide)
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- čas přidán 30. 06. 2024
- This is the ultimate calculus study guide for your university-level calculus and real analysis class. We will do 24 rigorous proofs for limits, including the epsilon-delta proofs, epsilon-N proofs (when x approaches infinity), M-delta (when the limit is infinity), and M-N proofs (when x goes to infinity and f(x) also goes to infinity) for limits. It's definitely the GPA saver that you need for college. Make sure you know the 4 big rigorous definitions for limits here: 👉 • 4 BIG limit definitions
File (public): 👉 / 24-rigorous-88483708
My hand-written notes for all these 24 proofs: (Patreons only) 👉 / notes-for-24-88670205
Join this channel to get access to the perks:
/ @blackpenredpen
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0:00 24 limit proofs with definitions
Q1. 0:29 epsilon-delta proof, linear case
Q2. 3:51 epsilon-delta proof, rational function
Q3. 12:04 epsilon-N proof, x approaching infinity, rational function
Q4. 17:18 M-delta proof, x approaching infinity, rational function
Q5 22:19 epsilon-N proof
Q6 28:34 epsilon-delta proof, quadratic case
Q7 32:26 M-N proof, square root function
Q8 36:06 epsilon-delta proof, square root function
Q9 42:47 εδ linear case
Q10 45:00 εδ limit proof, hard
Q11 52:21 limit of 1/x^2 as x goes to infinity
Q12 55:17 limit of sqrt(x-3) as x goes to infinity
Q13 58:24 limit of x^3 as x goes to 2
Q14 1:04:43 εδ linear case
Q15 1:06:40 εN proof
Q16 1:15:50
Q17 1:19:00
Q18 1:21:50
Q19: 1:26:46
Q20: 1:30:07
*start at 1:31:41, please change all the ε to M
Q21 1:32:24
Q22 1:36:42
Q23 1:41:21
*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
Q24 1:44:15 limit of x^2/(x+1) as x goes to infinity
*at 1:48:26, we need to add the condition that N is greater than 1. So please write N=max{1, 2ε}
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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
Very Good job
Another mistake.
You need to write N=max{ 1, 2M}
Not max{1 , 2epsilone}
:v
I'm missing some harder limits involving sin(x), e^x, log(x) etc. My class never covered that. Part 2 with those please
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
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
This video, like the "100 limits" and "100 derivates" are good asf, you're a savior teacher!!!!!! Thanks a lot
4 BIG definitions -> czcams.com/video/X3F2o_9qe1Q/video.htmlsi=wxxnnbzzvW_btnvk
video is private D:
Jesus loves you ❤️ please turn to him and repent before it's too late. The end times described in the Bible are already happening in the world.
lol@@L17_8
please to see
Thank you for doing this channel. You make calculus fun and enjoyable.
Lets goooooo. The school just started so I can flex with this knowledge. 😂😂❤❤
Be cautious people may start calling u a nerd
@@user-ex7fq9dy5e I'm already called a nerd sometimes so I don't really care
Here in India , if you are a topper , everyone respects you.
Thanks professor!!! Fortunately now since it's September as maths channels we'll get more views!!! Thanks a lot I love your videos!!!🥳🎊
I really liked the video. I learned a lot on your channel. Continue with your wonderful work.
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
Bro you are the goat for saving me I was struggling with epilson delta and now I have finally grapsed the concept
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
Wonderful lecture. May you have good health for all the best.
Thanks, bprp. Those proofs are really important in Calculus, mainly for deeper students or math (Uni) students. Gonna revise from time to time.
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
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.
This was so effective
Thank you so much
This video was so so so helpful. I only wish it existed back in 2015 when I was taking Real Analysis.
THANK YOU SOOO MUCH I WATCHED ALL THE WAY THROUGH 😍😍😍😍❤❤❤❤
I was struggling epsilon delta finally got it thanks
Thanks for your work.
Thanks a ton my friend, watched the whole way through😃.
watching still at 3:26 am (preparation for calc 1 midterm) thanks a lot for this video!
You know you're a nerd when this video drops and you audibly say "hell yeah!"
I have an exam in a week 😭...you uploaded it at the perfect timing
Hi bprp, I wonder if this course will also cover Cauchy sequences and continuity, especially with multivariable functions?
SUPERBBBBBBBBBB!!! what books do you recommend for more explanation and exercises on real analysis? Thank you!!
you just saved my year
On question 15 you could solve it too by factoring x^2 - 1 so it would be x/(x+1)(x-1)
thank you so much sir
Please teach integration by graph and do also teach all graph like hyperbola graph of mode etc
From 1:31:41 to 1:32:10 (problem 20) replace epsilon with M everywhere.
Thank you, I was wondering about that
Yes. I mixed that up. Thanks!
@@blackpenredpen Btw, I kept watching. Thank you, it was a wonderful practice for my real analysis course.
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 :-)
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.
you are very intelligent person
For the last question, choosing N=2M allows for 01). I love your videos, keep up the great work❤
Ah thanks!! Let’s do N=max{1, 2M} so that >1 is more clear
@@blackpenredpenWhy do we choose max instead of min? Nice video!
god im STOKED for this
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?
use L'H twice, but use multiplication law in 2nd one as well
@teddygaming4076 fair enough I guess without lh would be what I was intending.
#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.
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?
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...
Day 1: 100 Differential equations please. You are the greatest teacher
thank you. after this I decided to take analysis lol
respect brother
Thanks.
you are the greatest 🥰🥰🥰🥰🥰🥰
You are fantastic
1:02:27: I was wondering if from the fact |x-2|
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?
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.
Thanks
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
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 CZcams videos on calculus.
genial bro , mas videos asi porfavor de todo el cálculo. thanks
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)
the same for number 24
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
WE LIT!!!!
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?
Q19:
Can N = 1/ ε
Instead of N = 1/sqrt(ε) ?
Yes. That’s fine too.
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
Could u solve this one
4x2-3x+2/8x2-6x+1
=1/2
N & ε case
Thank u
I am still watchin!
Awesome!
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.
Hello professor, please make 1 video for examples in which delta epsilon defination fails because the limit donot exist there.
Do you have a specific limit in mind?
@@blackpenredpen limit of sin(1/x) at 0.
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.
■
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?
Are \infty and +\infty the same? I think no. So in #3 we have to suppose that |x| > N, aren't we?
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?
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
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
hi bprp
try to give an example of a limit that can't be proven
Yes!
53:39 N has to be bigger than 1, not zero, for x2 to be bigger than N2.
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.
@@blackpenredpenYou are right, forget what I said.
You should find the derivative of x!
LETS FUCKING GOOOOOOO
12:06
Don't mind me, I'm just marking where I left off
1:23:51
Pb 20
Should it not have been >M
Could u please look at it again
Thank u
what about proofs for limits that don't exist?
Limits are cool ✌️✌️😎
Why d owe need to put min as 1
Do you know how to rebuild a motorcycle engine? Restore a vintage audio amplifier? Oil painting or play the bass? Downhill ski?
Why do we have the minimum for delta as 1?
#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?
yes got it too
55:12
Can you do a proof of a limit that doesnt exist please?
11:41
Why not just take the slope at that point and take delta
The derivative is based on the limit existing at that point, and this is what you’re trying to show (rigorously).
On the last question couldnt you just drop the x2
Lim
bprp->24 =🏆🏅🏅🤗
Hi
At q 15 i go √1/epsilon +1
Pb 24
Why is x>1
(1) 2x-3 is continous function and then lim=function. Il resto è fatica sprecata 😂
❤❤
what level math is this
High school AP calculus or first year university calculus 1.
Maybe it’s just me, but I understand how the questions work but have no clue how it’s proving anything 😂
i was working along with e^itheta formula, and i just had gotten a crazy idea, replacing theta with i so it becomes e^i^2 = cos i + isin i . 1/e = cos i + isin i and i found out complex trigonometric functions exist. when i derived the value of cos i, I found it is equal to cosh 1. I have not learned about hyperbolic functions. Could anyone say how this happened, and if bprp sees this please respond
Explanation is basically just definitions, by definition, cos(ix) = cosh(x), so setting x = 1 yields cos(i) = cosh(1) = cosh(-1) due to it being an even function
Alternatively, many other ways to derive this can be found e.g.
cos(i) = e^-1 - isin(i) which (by definition of sinh(x)) = e^-1 + sinh(1). As sinh is an odd function, sinh(1) = -sinh(-1) therefore cos(i) = e^-1 - sinh(-1). e^x-sinh(x) = cosh(x) therefore cos(i)=cosh(-1)=cosh(1)
@@adlg5158 yeah thanks a lot. I checked hyperbolic functions in google and found this out and found the meaning and variations formulas using hyperbolic functions. I commented this thinking i found something new but then it turned out i just commented something which already exists but thanks for the explanation
write Q.E.D.
Limits are "hard" to calculate in singularities... elsewhere lim=function
😂😂😂😂😂
This is a great example of how NOT to learn math. 😂 This is not what math is about, this is just showing the mechanics and "monkeying" through problems. But NOT showing why these choices are being made or what some the terms even are. Math is the "why" and the thinking about how to solve problems. And not so much getting an answer.