epsilon-N definition for a limit at infinity (introduction & proof writing tips)
ฝัง
- เผยแพร่เมื่อ 3 ส.ค. 2024
- 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
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Check out the εδ definition ultimate introduction 👉 th-cam.com/video/DdtEQk_DHQs/w-d-xo.html
Right 👍👍
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.
Thank you for making this demonstration waaay easier than what I learnt in school. Time to flex it on my teachers 😜
This is just what I studied for an upcoming exam, great to refresh my memory. Thanks 🙏
First time I understand. Great explanation. Thanks
Good that you gave this video I'm about to start limits in my calc course
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.
Man, you bring out the inner math genius in me, and I'm 62. I follow your logic perfectly
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”.
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.
@@yanceyward3689to be exact, for all n > N that has to satisfy, otherwise it's a limit point
Damn, I appreciate your videos even more given my professor couldn't explain it properly in 3 hours.
Welcome back man!
Very Very Thank you sir ❤
Thank you!
Could you do a video on how to do the proof backwards/both ways?
Nice video
To infinity and beyond 🚀🥳🤸♥️💫
beautiful
L'hospital: it that even a question?
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!
respect from bmstu
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
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
@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
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
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?
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.
@@asifthatwouldeverhappen thanks
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
still appealing to the version in English
How to do it with negative infinity
hey blackpenredpen, can you solve
x = i^x
as in, an infinite power tower of i's.
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.
Hi, what do you do if you have a minus sign in the denominator, so you can't get rid of the absolute value??
a minus sign in the denominator is just a minus in the numerator, no difference
Do you have to use the max?
What’s wrong with N being negative?
"approaches infinity"...
@@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
Did you get the answer to your question? Cause I have the same doubt
@@stlegendff7390 nope unfortunately not
When I will have access to America?
Can we not do taking 1/x tend to 0 if x tends to infinity
Not calculate but prove it is 2/3. So the limit definition is applicable.
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)
@@Ninja20704 thanks brother
Don't we need to write the conclusion at the end?
3:36
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... 😎
I just multiply the top and bottom by (1/x).
Are you not allowed to use L’Hopital for a prove?
Of course not. That's not a proof, that's a calculation technique. Furthemore, these easy limits are usually covered before derivatives.
@@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
@@photophone5574 it's allowed if you had previously proved l'Hospital's rule. Which I don't think is the case.
No you definably can’t due to the lack of proof and evidence
please stop bringing this overly used method up, notably whose validity is beyond your question. Utter tiresome.🙄
Lhlpital 3secs
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)?
is it 3 ?
This chanels name starts with my favourite k pop bands name.
the one who copy la campanella?
dalil d La Hospital
Undefined : Proof Theory Graph Trayektory
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
Use L'Hopital's rule what's the problem bruh😅
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
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?