epsilon-N definition for a limit at infinity (introduction & proof writing tips)

Рет қаралды 52,002

Күн бұрын

I will introduce the epsilon-N definition of a limit and will also show you how to write a rigorous proof for a limit as x goes to infinity. I will help you understand the definition by using a specific epsilon value, graphs of the function, and the line y=L-epsilon, and solving for a corresponding N. And remember "given, choose, suppose, and check".
#calculus #math #maths #college #blackpenredpen
This is similar to the epsilon-delta proofs for limits and you can check out that introduction here: 👉 • epsilon-delta definiti...
0:00 I will help you understand the εN definition for a finite limit at infinity
0:19 the 4 main cases of a rigorous definition of a limit
2:24 the εN definition
4:14 how to easily write a rigorous limit proof
11:32 an actual example with ε=0.02 and find a N
#calculus #blackpenredpen #realanalysis #math #college
🛍 Shop my math t-shirt & hoodies: amzn.to/3qBeuw6
----------------------------------------
💪 Support the channel and get featured in the video description by becoming a patron: / blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall Andr3w11235 Cadentato Joe Wisniewski Eric Maximilian Mecke
----------------------------------------
Thank you all!

Пікірлер: 75

@blackpenredpen 10 ай бұрын

Check out the εδ definition ultimate introduction 👉 kzfaq.info/get/bejne/esqkeLScxKqsgac.html

@thatomofolo452 10 ай бұрын

Right 👍👍

@scottleung9587 10 ай бұрын

This was my absolute worst nightmare while taking Modern Analysis in college! But many thanks for walking us thru the steps of how this proof came to be.

@shiftsync9988 10 ай бұрын

This is just what I studied for an upcoming exam, great to refresh my memory. Thanks 🙏

@varun3282 10 ай бұрын

Good that you gave this video I'm about to start limits in my calc course

@yassinelaouadi305 10 ай бұрын

Thank you for making this demonstration waaay easier than what I learnt in school. Time to flex it on my teachers 😜

@haasjeoverkonijn6961 4 ай бұрын

First time I understand. Great explanation. Thanks

@lawrencejelsma8118 10 ай бұрын

As an engineer working on engineering precision mathematics we find the x equivalent in like time measurements all the time when 2x + 1 approximates to 2x with no significant error in precision. 3x + 4 would need a larger x to equal the same low error approximation as 2x + 1 approximate 2x replaced as 3x + 4 approximately is 3x for x >> 1 for large x. For example, x = 10^4 then the function approximates to 2x/(3x) or x in numerator and x in denominator show we are so close to 2/3 no matter if you have a ripple 1 in the numerator and a ripple 4 in the denominator compared to the 2x value / 3x value ... Not the 2 value / 3 value limit mistake in thinking.

@ItsMeTheUser 10 ай бұрын

Welcome back man!

@dmitrybak9372 10 ай бұрын

For epsilon-delta proofs, when speaking through the proof out-loud, it becomes way more obvious for the student if the speaker says not just “for all epsilon greater than zero, …”, but “for all epsilon, no matter how small you decide to choose it, …” - i.e. emphasizing that our intent is to “make” epsilon “smaller and smaller”. Also “given arbitrary epsilon greater than zero” -> “given arbitrary small epsilon greater than zero”.

@yanceyward3689 10 ай бұрын

This is an important point that often gets lost in these discussions- you can make epsilon as small as you like and will always be able to find a delta or N that satisfies the delta/N inequality, and vice versa, if the limit is in fact L. If the L is not the limit, then you will get a contradiction.

@nayjer2576 3 ай бұрын

@@yanceyward3689to be exact, for all n > N that has to satisfy, otherwise it's a limit point

@hritamkashyap 4 ай бұрын

Very Very Thank you sir ❤

@nonentity168 9 ай бұрын

Damn, I appreciate your videos even more given my professor couldn't explain it properly in 3 hours.

@BusyBlueLion 8 ай бұрын

Thank you!

@etcthedeimos 10 ай бұрын

Nice video

@helphowdoinputusername3571 10 ай бұрын

Could you do a video on how to do the proof backwards/both ways?

@basilbrush7878 3 ай бұрын

Man, you bring out the inner math genius in me, and I'm 62. I follow your logic perfectly

@thatomofolo452 10 ай бұрын

To infinity and beyond 🚀🥳🤸♥️💫

@user-nw6cw7vf8r 10 ай бұрын

L'hospital: it that even a question?

@IloveTwinkies412 8 ай бұрын

respect from bmstu

@teddy05p 9 ай бұрын

Lovely video, though I would like to point out a correction that for epsilon-N it should be stated as a defined sequence lets say a_n, so rather a set of a sequence than as you said a function:) But both work fine I guess! Ty for the videos!

@Ninja20704 10 ай бұрын

My teacher gave us some simple limit proof qns in my proofs class when teaching us proof by construction. And this was one of them. Quick question, if we wanted to prove a limit as x->-inf, do we just change to N

@CatchyCauchy 10 ай бұрын

I think there are many ways to define it for example: lim x-> -inftx f(x) := lim x-> infty f(-x) if it exists Or using sequences: For all sequences (xn)n with xn -> -infty, then we say lim x->-infty f(x) exists if and only if lim n->infty f(x_n) exists and is the same for all sequences I think yours is probably aquivalent

@Xorven2 10 ай бұрын

@Ninja20704 Yes. A usual definition of the existence of a finite limit L of a real function f at -∞ is : For all Ɛ>0, it exists N

@user-qj3rv2mo1b 5 ай бұрын

I have seen an example where the δ chosen was greater than ε. I was wondering would it not throw δ outside the ε window. Can the chosen δ ever be greater than ε? Thank u

@sadi_supercell2132 Ай бұрын

How to do it with negative infinity

@broytingaravsol 10 ай бұрын

still appealing to the version in English

@mihailangelov6 10 ай бұрын

When I will have access to America?

@ziro4614 10 ай бұрын

I asked myself what the limit of (1+1/x)^(1/x) when x goes to infinity. I think it goes to 1 but I don't have a way to show it and when I asked Wolfram Aloha it says 1 but the Step-by-step solution kinda goes like e^(0/infinity) equals 1, which it say is the solution. Can someone help me?

@asifthatwouldeverhappen 10 ай бұрын

Write the expression as exp((1/x)*ln(1+1/x)) As x tends to infinity, 1/x tends to 0 and ln(1+1/x) also tends to 0. Therefore, the limit is equal to exp(0*0) = e^0 = 1 You can then write the proof it in the epsilon N form.

@ziro4614 10 ай бұрын

@@asifthatwouldeverhappen thanks

@Ninja20704 10 ай бұрын

A simpler way to calculate the limit is that 1/x goes to 0 as x goes to infinity. So the base approaches 1 and the power approaches 0. This is not an indeterminate form, so we can legitamately conclude that the limit is 1^0=1

@Shaan_Suri 5 ай бұрын

Hi, what do you do if you have a minus sign in the denominator, so you can't get rid of the absolute value??

@cjjk9142 Ай бұрын

a minus sign in the denominator is just a minus in the numerator, no difference

@Happy_Abe 10 ай бұрын

Do you have to use the max? What’s wrong with N being negative?

@spiderjerusalem4009 10 ай бұрын

"approaches infinity"...

@Happy_Abe 10 ай бұрын

@@spiderjerusalem4009 of course, but I’m saying if such a small N works that it’s negative should still be fine if the inequalities hold, same way N can be a random small positive real number that isn’t close to infinity

@stlegendff7390 8 ай бұрын

Did you get the answer to your question? Cause I have the same doubt

@Happy_Abe 8 ай бұрын

@@stlegendff7390 nope unfortunately not

@Douae1111 8 ай бұрын

3:36

@thebestchemicalelement4455 10 ай бұрын

hey blackpenredpen, can you solve x = i^x as in, an infinite power tower of i's.

@nickulman9739 10 ай бұрын

i^i^i^…..=x x=i^x x=e^(ln(i^x)) x=e^(xln(i)) note: (ln(i)= (pi*i)/2 for the complex logarithm principal branch, this can be observed also through rulers identity e^(pi*i) = -1 as putting both sides to the power of 1/2 results in e^((pi*i)/2)=i.) x=e^(x*i*pi/2) x/(e^(x*i*pi/2)) = 1 x(e^(-x*i*pi/2))=1 x*(-i*pi/2)*(e^(-x*i*pi/2))=(-i*pi/2) W(x*(-i*pi/2)*(e^(-x*i*pi/2)))=W(-i*pi/2) -x*i*pi/2=W(-i*pi/2) x=(2i/pi)*W(-i*pi/2) Sorry for bad formatting, am commenting on phone.

@neekunjchaturvedi 10 ай бұрын

Can we not do taking 1/x tend to 0 if x tends to infinity

@sivanaidoo5602 10 ай бұрын

Not calculate but prove it is 2/3. So the limit definition is applicable.

@Ninja20704 10 ай бұрын

You can use this definition to prove that. Given any epsilon>0 Choose N = 1/epsilon Suppose x>N. Check |1/x-0|=1/x (x>0 so 1/x>0)

@neekunjchaturvedi 10 ай бұрын

@@Ninja20704 thanks brother

@MikiMiusA 6 ай бұрын

Don't we need to write the conclusion at the end?

@jotajaviergonzalezgarcia7504 10 ай бұрын

Lhlpital 3secs

@JB-ym4up 10 ай бұрын

I just multiply the top and bottom by (1/x).

@SamsungA04e-dp7kj 3 ай бұрын

dalil d La Hospital Undefined : Proof Theory Graph Trayektory

@ceo1OO 9 ай бұрын

The blackPenredPen guy is putting on weight: ... he must be eating alot of general Tso's chicken... lol.. it's alright, I eat general Tso's chicken also... 😎

@AdityaKumar-gv4dj 10 ай бұрын

Please solve this question for me: Let f(x)=(x²+x+1)/(x²-x+1), then the largest value of f(x) for all x belongs to [-1,3]. Can you please teach me how to apply x's bound over such f(x)?

@arkodasgupta0412 7 ай бұрын

is it 3 ?

@naxnusternann332 10 ай бұрын

Are you not allowed to use L’Hopital for a prove?

@SimsHacks 10 ай бұрын

Of course not. That's not a proof, that's a calculation technique. Furthemore, these easy limits are usually covered before derivatives.

@photophone5574 10 ай бұрын

@@SimsHacksI think you are right on it coming before derivatives, but calculation techniques do work in proofs. Calculation techniques in proofs is just called algebra

@SimsHacks 10 ай бұрын

@@photophone5574 it's allowed if you had previously proved l'Hospital's rule. Which I don't think is the case.

@zarkuz8702 10 ай бұрын

No you definably can’t due to the lack of proof and evidence

@spiderjerusalem4009 10 ай бұрын

please stop bringing this overly used method up, notably whose validity is beyond your question. Utter tiresome.🙄

@siamchowdhury3987 10 ай бұрын

This chanels name starts with my favourite k pop bands name.

@picup30296 10 ай бұрын

the one who copy la campanella?

@ingramr88 10 ай бұрын

If the first thing you comment it L’Hospitals, we already know that you’re not going to get out of this video what you need to get out of this video. It’s very telling. Come back after a discrete math class then look at this video lol

@ussr8410 10 ай бұрын

Use L'Hopital's rule what's the problem bruh😅

@tiangong6616 10 ай бұрын

But this is about how to write a formal rigorous proof, not obtain the final answer, L'Hopital's rule is more like a calculation technique

@willthecat3861 10 ай бұрын

My cat, and I, have watched a lot of BR's proof vids. My cat is much better than me at giving all the proofs... just copies BR's proof. I don't think my cat understands math... do you?