Epsilon delta limit (Example 4): Limits at infinity
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- Опубліковано 7 вер 2024
- This part of the epsilon-delta series covers limits at infinity. You can find Examples 1 and 2 on blackpenredpen's channel, and Example 3 on my channel. Enjoy!
Note: I realized after the fact that this limit may be a bit too simple, but if you want to prove that the limit of f at infinity is L, you do the same argument as in the video, except with |f(x) - L| instead of just |f(x)|.
you can tell this guy really loves his job
Dr. Peyam: I am a maths teacher and I hope I can bring even half the joy you show in these videos to my own classes. You inspire me. Thank you.
The most positive teacher I have ever seen, come teach at my school, I beg you!!!!!!!!!!!!! WE NEED YOU!!!!!!!!!
I am beyond bummed out that I didn’t discover this channel until after I graduated from undergrad. I would have been a much happier and successful student. Thank you for the content!!
I have been trying to understand this formal definition of limit concept. you made it so easy. Thank you
You are a great man doctor. May you prosper and be blessed by God himself. Thank you so much.
I will forever believe that these proofs are absolutely trivial, although that's not a bad thing, it just confirmes that the idea of the limit is as intuitive as it can get. So I have fixed feelings about them!!
omggg i finally understand. in my calc class my teacher used uppercase D and E to illustrate N which confused me so much bcuz i didnt understand how delta and epsilon connected to them (specifically, why different variables were needed) THANK U SO MUCHHHH 😍😍😍😍
Thank u so much sir 🙏....your explanation made it very interesting and easy👍
I think the reason people have trouble with these problems is the proofs are circular, sort of like circular reasoning based on the definition.
I went to ProfRob on UA-cam and he explained what was going on along the way in a couple simple examples.
Yeah that is not the case at all. It is less about circular reasoning and more about the concept of SCRATCHWORK vs PROOF. It is like proving a function is surjective. You end up doing a bunch of scratchwork before writing down the final proof so it seems like circular reasoning. Just read the final proof.
Ohh wait....I was studying something 😍😇😇😇....Just find your channel .... Sir! You was so awesome !!.....Will watch more as required 😇😇
So flamboyant
Dr. Payem you are outstanding and explained very well, thank you. Could u also use graphs of the function and show epsilon, delta, M and N on it to make it more visual.
Thank u
may the dear Lord bless you sir you made this topic so easy ah
I didn't want this solution but my question was so related to this one. Thank you so much. And I must say be happy always like this :)
I really like your content... and your voice ;)
Wow interesting lecture
This is absolutely great!
bless you my lord
My first-year calculus professor said that limit proofs made him cry literal tears in his first-year 😂
Dr. Peyam: Please proof Lim (x-->0) e^X=1 with epsilon-delta.
I get how it works but the only thing that still holds me is the f(x)>M or f(x)
Prove that the lim(x/x^2+1)=0 at x goes to negative infinity please you answer it by epsilon delta definition
Love the arrows
You are so cool.
in the beginning would you not write x>sqrt of 1/epsilon rather than x>1/sqrt of epsilon or do I need to brush up on my algebra
Actually I think in India teacher s teach so seriously maths that when he tried to become interesting and cool
It seems fool
But good
Exactly
It is very helpful
thank you
Hello, i'm confused. I understand your demo, but, don't you have to start your proof from |x - infinite| > delta (or N) implies |1/x²| < epsilon ? I don't understand how you get rid of the infinite in the absolute value.
When x goes to infinity instead of approaching a finite value the rule changes from |x - L| < delta.
This is because infinity is not a number and we want to make x arbitrarily large.
Here is the modification:
There exists a delta such that x>delta implies |1/x^2|
Why is that the modulus of 1/x^2 is less than the epsilon ? And can I know what is actually the epsilon meant?
epsilon is just an arbitrary number that we pick, usually a really small number.
Why this definition asks for all Epsilon greater than zero. If Epsilon is the rate of variation cannot be negative or am I wrong?
Epsilon is an error. There are no negative errors
@@drpeyam Thanks.
I can't go home very happy because I'm already at home very happy :)
what if we guess wrong ?
Yess
yass
What would happen to Part 2 if we also needed x>4 alongside N = 1/sqrt(epsilon)?
Would we then "let 0 < epsilon < 1/16 instead of writing let epsilon > 0" as we have a condition that x>4, which means that epsilon < 1/(4^2) = 1/16.
I'm asking this question because in some cases, to simplify we need to restrict the domain of "x" a little bit. Thanks in advance!
Indian Drake teaching me math
Hahahaha
Very happy :).
awesome
I was once exposed to a recursive definition for a sequence called Q. For x < 3, Q(x) = 1, for x >= 3 then Q(x) = Q[x - Q(x-1)] + Q[x - Q(x-2)]. I have observed many interesting properties regarding this sequence, but it is very frustrating to try and prove anything mathematically because it is soooo recursive - lol Can anybody help me out :-)
I have been looking at a notion of derivative where you examine the difference between Q(x + 1) and Q(x) = Q(x + 1) - Q(x)
Here I see a very chaotic and almost fractal graph (the original Q is also fractile in nature) with a slope near to 1/2 on its maxima.
Note: Using this form of derivative puts the graph centered on the X-axis which is sort of nice to me.
So Q begins like this: 1, 1, 2, 3, 3, 4, 5, 5, 6, 6, 6... and dQ looks like: 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, ...
In an Excel spreadsheet you can use the formula: = INDEX(B$2:B3, A4 - INDEX(B$2:B3, A4 - 1)) + INDEX(B$2:B3, A4 - INDEX(B$2:B3, A4 - 2)) in a column that begins with a pair of ones to generate the sequence. It is (so far for me) an undecidable proposition that one must only retain the last half of the sequence in memory in order to generate further numbers.
Also, note that I have the row indexes in column A when using the Excel formula above starting in row two with the formula: = row() - 1
P.S.
Sorry for hijacking this thread, but I was watching this while contemplating the Q issue - lol
Oh.. Here is a png file of the first 1476 itterations of dQ: www.dropbox.com/s/q6vub94mcedxuy8/dQ.PNG?dl=0
The only mathematicians that I know of who are interested in this sequence are the professor who showed it to me decades ago, and myself ;-)
These kind of rascals do not seem to be so amenable to calculus. This fascinates me.
Well, using the chain rule, Q'(x) = [Q(x - Q(x - 1)) + Q(x - Q(x - 2))]' = Q'(x - Q(x - 1)) (1 - Q'(x - 1)) + Q'(x - Q(x - 2)) (1 - Q'(x - 2)), giving you a recursive definition for the derivative.
*This comment is incorrect*
But wouldn't the chain rule demand an infinity long string (actually two of them separated by a plus sign)?
This is a sequence. There is no derivative, so the chain rule doesn't apply. What you are referring to is the difference of successive terms. If you want, I can make a video studying the sequence. Thanks.
I did some more thinking about sequences last night and how I use them in discrete systems. I work with digital radio systems using (or soon to be using) QAM256. But basically, there is an analog to digital converter that samples at a high rate (and as tech evolves this always get even faster). So what my software gets to 'look' at is an array of numbers.
You say that sequences do not have a derivative, but I will show how they kind of do.
Of course, we are all taught that the derivative of f(x) is equal to the limit as dx goes to zero of: (f(x) - f(x + dx))/dx
So what is a reasonable approach when dealing with a sequence of numbers. Here is my idea: The 'distance' from one number to the next is always one (I stole this idea from my physics friends who set the speed of light to 1).
So, if the limit is forced to one, then the whole thing becomes a simple object. The derivative of a sequence becomes a simple subtraction operation: f(x) - f(x + 1), and this is the one I used in my Excel pdf image that I published. Again I like laying it on its side (so to speak) in this way because the rendering engines work better in Excel and similar software when it is flattened out.
I do appreciate your involvement in this inquiry.
And here I have explained how to take a kind of derivative of an arbitrary sequence of numbers.
God bless.
what is epsilon
where we use the epsilon delta definition in math i mean how we use that thing
To prove limits rigorously. It also sets up a framework in your mind how rigorous proofs work.
yes
I wanna study in this guys class...
I found N within my couch. It was edible
Peyan=peyam in Tamil translation ur name
Really , don,t get it...
Yeaaahhhh😀😀😀😂👍
lol. silly but smart