epsilon-N definition for a limit at infinity (introduction & how to write the proof)

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

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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Thank you all!

[scottleung]

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.

[shiftsync]

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

[radkiller]

you just explained this so well, ive watched so many videos but yours made me clearly understand this. Thankyou so much!!!

[yassinelaouadi]

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

[haasjeoverkonijn]

First time I understand. Great explanation. Thanks

[lawrencejelsma]

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.

[dmitrybak]

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”.

[varun]

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

[noonstar]

Thank you so much these videos save my life. But do you have a video on if there's an M to find instead of an N?

[helphowdoinputusername]

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

[Ninja]

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<0 and x<N?

[nonentity]

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

[basilbrush]

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

[WritersDigest-bf]

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

[abhinnkaushik]

is it necessary to speify the domain of N? like N is s subset of Real numbers when we complete the proof?

[teddyp]

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!

[Shaan_Suri]

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

[thebestchemicalelement]

hey blackpenredpen, can you solve

x = i^x

as in, an infinite power tower of i's.

[ziro]

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?

[Happy_Abe]

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