Real Analysis 26 | Limits of Functions

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This is my video series about Real Analysis. We talk about sequences, series, continuous functions, differentiable functions, and integral. I hope that it will help everyone who wants to learn about it.

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00:00 Intro
00:59 Definition
05:50 1st Example
06:29 2nd Example (Polynomial)
08:12 Credits

#RealAnalysis
#Mathematics
#Calculus
#LearnMath
#Integrals
#Derivatives

I hope that this helps students, pupils and others. Have fun!

(This explanation fits to lectures for students in their first and second year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)

[Hold_it]

You really make all of this seem so easy. Keep up the good work! :D

[sharonnuri]

Your explanation better describes the relation between limit of sequences and limit of functions. Thats the advantage of watching your videos compared to a textbook :)

[skapun]

You should make videos on Proof Exercises from different textbooks!

[id-icou]

In France, we have a slightly different definition for the limit of a function, we in fact allow the sequence to take the value x0 (i.e we do not have the “0<” in 0<|x-x0|<δ of the ε-δ definition). So in France, the function of example (a) doesn’t have a limit at 0. And our definition of continuity is simply that a function is continuous at a point (in the domain of the function) if it has limit at that point (in this case the limit is automatically equal to the value of the function at that point)

[dhn]

2:05 Does x_0 have to be in I ? For example the function f(x)=sin(x)/x maps I=R\{0} to R. The point x_0=0 is not in I, but we still have lim_{x->0}sin(x)/x=1

[blue_infinity]

This is a great lecture! Thank you so much.

[gyanprakash]

I just subscribed your channel. You make amazing videos, keep up the good work!!

[liberated]

damn, i learnt this material so many times yet this explanation made it seems like new subject.

[jukkejukke]

Could you please give an example of a series? I mean a series is based on natural numbers as inputs, but inputs to a function on an interval are real numbers.

[MrOvipare]

So we have seen that every polynomial must be continuous. Can we say that ANY continuous function is representable with a polynomial?
This is basically the idea of taylor expansion, but is it rigorously always true? If not what are the limits? I'm aware that taylor expansion is always "local" (centered on x0) so this is probably a reflection of the fact that continuousness is intrinsically a local idea.

[lucasguarracino]

In 3:46, when you say that we assume that there exists at least one sequence contained in I-{x_0} that converges to x_0 (or equivalently that x_0 is not an isolated point), you don't write it. I think it is important it to be written in the definition of limit you make, because if one just read that definition, the limit when x tends to an isolated poin of I is defined and could be any real number! Or are you supposing that I is an interval and I missed that part? Great video, thank you!

[ahmedamr]

Brilliant video as usual!
I stop at 5:00, I look at the definition of continuity and a couple of questions come up:
1) Since any sequence is infinite, and x_0 is the limit point, the exclusion of the limit point x_0 from I in the second line follows directly. This exclusion therefore is explicitly written only for clarity. Is that correct?
2) I look at the definition and at no point do I see f(x_0) = c. Shouldn't that be written at the start perhaps?

[angelmendez-rivera]

So in summary, if:

0. f is a function from I to R, where I is an open interval of R.

1. X is an element of I.

2. There exists some sequence x : N —> I\{X} that converges to X.

...then we say that, lim f(x) (x —> X) = c if and only if lim f[x(n)] (n —> ♾) = c for every sequence x : N —> I\{X} that converges to X.

Furthermore, f is continuous at X if and only if c = f(X).

[HungDuong-dtlg]

This is a great lecture! Thank you so much!