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
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
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
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.
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
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!
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
I hope you read my comment sir, Day after tmro I have my real analysis exams, watched the video fully from start to finish and understood everything , really appreciate your efforts sir,Thank you!!!
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?
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 :-)
#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.
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.
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.
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?
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
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
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...
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?
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
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.
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. ■
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)
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
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 RUclips videos on calculus.
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
Trust me, they're too hard. Use sandwich theorem for them, it's more pleasant
sorry I'm not a pussy to say something basic is "too hard"
@@LordBrainz sandwich theorem is to find the limits not to prove them
@@GaryMahal The squeeze theorem: Let A be a subset of R, let f: A -> R and let c be an element of R. Further, let c be a cluster point of A. If a
This video, like the "100 limits" and "100 derivates" are good asf, you're a savior teacher!!!!!! Thanks a lot
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
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 😊
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.
Lets goooooo. The school just started so I can flex with this knowledge. 😂😂❤❤
Be cautious people may start calling u a nerd
@@CKNGAI-r8x I'm already called a nerd sometimes so I don't really care
Here in India , if you are a topper , everyone respects you.
😂
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
Thank you for doing this channel. You make calculus fun and enjoyable.
Thanks professor!!! Fortunately now since it's September as maths channels we'll get more views!!! Thanks a lot I love your videos!!!🥳🎊
I am so grateful for what you do
Bro you are the goat for saving me I was struggling with epilson delta and now I have finally grapsed the concept
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.
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!
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
the best teacher i ever seen
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
Thank you very much! I have a test this Friday and this video really helps me a lot !!!
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.
You know you're a nerd when this video drops and you audibly say "hell yeah!"
i finally understand delta epsilon proofs. ur amazing.
I hope you read my comment sir, Day after tmro I have my real analysis exams, watched the video fully from start to finish and understood everything , really appreciate your efforts sir,Thank you!!!
This video was so so so helpful. I only wish it existed back in 2015 when I was taking Real Analysis.
I was struggling epsilon delta finally got it thanks
This was so effective
Thank you so much
Thanks for your work.
been watching through this all the from start to end!!!! it was really fun may be do more challenging problems by using other real valued functions 👀
THANK YOU SOOO MUCH I WATCHED ALL THE WAY THROUGH 😍😍😍😍❤❤❤❤
I have an exam in a week 😭...you uploaded it at the perfect timing
Thank you for the great video!
watching still at 3:26 am (preparation for calc 1 midterm) thanks a lot for this video!
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.
how did we correlate delta to M at 21:35 😭
Thanks a ton my friend, watched the whole way through😃.
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 :-)
that is great ,keep it up!!!!!!!!!!!!!!!!!,thank you
#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.
you just saved my year
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
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?
you are very intelligent person
you are the greatest 🥰🥰🥰🥰🥰🥰
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.
thank you so much sir
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!
SUPERBBBBBBBBBB!!! what books do you recommend for more explanation and exercises on real analysis? Thank you!!
respect brother
Thanks.
damn if i knew about this channel earlier.still better late than never
thank you!
On question 15 you could solve it too by factoring x^2 - 1 so it would be x/(x+1)(x-1)
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.
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?
Please teach integration by graph and do also teach all graph like hyperbola graph of mode etc
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.
god im STOKED for this
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?
1:23:51
12:06
Don't mind me, I'm just marking where I left off
You are fantastic
Day 1: 100 Differential equations please. You are the greatest teacher
Q19:
Can N = 1/ ε
Instead of N = 1/sqrt(ε) ?
Yes. That’s fine too.
1:02:27: I was wondering if from the fact |x-2|
Hi bprp, I wonder if this course will also cover Cauchy sequences and continuity, especially with multivariable functions?
Why d owe need to put min as 1
Thanks
Lim
bprp->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
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
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
thank you. after this I decided to take analysis lol
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...
I am still watchin!
Awesome!
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?
Pb 20
Should it not have been >M
Could u please look at it again
Thank u
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
in example 13 why not take |(x+2)|^2< 25 and take just for x?
Limits are cool ✌️✌️😎
just wrote my calc midterm, studied proofs for over 20 hours in the last three days.
there were no proofs on the exam.
11:41
I watched till end
genial bro , mas videos asi porfavor de todo el cálculo. thanks
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.
what about proofs for limits that don't exist?
55:12
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.
■
Could u solve this one
4x2-3x+2/8x2-6x+1
=1/2
N & ε case
Thank u
hi bprp
try to give an example of a limit that can't be proven
Are \infty and +\infty the same? I think no. So in #3 we have to suppose that |x| > N, aren't we?
Why do we have the minimum for delta as 1?
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
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
On the last question couldnt you just drop the x2
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 RUclips videos on calculus.
Can Delta=4 epsilon be an answer
Can you do a proof of a limit that doesnt exist please?
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).
Do you know how to rebuild a motorcycle engine? Restore a vintage audio amplifier? Oil painting or play the bass? Downhill ski?
Very tope learning I am idmaier you