2.16 Limits and composition of functions

There isn't a nice law for limits of compositions of functions: knowing that two functions have limits does not tell me whether the composition has a limit. Why? And what can we do instead?

Related videos:

0:00 Introduction
1:10 False composition theorem
1:46 Composition continuity
2:02 Counter-example
3:42 Core of the problem
5:59 Fixing the theorem
7:23 Final notes

[ebrahimmohamud]

Might be the most counterintuitive theorem I have seen so far but after watching this many times it finally clicked and it makes perfect sense. This is really valuable info so thanks a ton!!

[ll-bcgn]

7:02 what does it even mean by "f(x) cannot be L for x on an interval centered at a when the limit of f close to a is L"? Isn't this countering the definition of limit. f(x) can be L. Correct me if I am wrong. (or is this the case that we can apply squeeze theorem?)

[invincible]

sir u are just amazing, u provide intuition as well as proofs in your lecture in a very elaborative way .THNKU SO MUCH SIRRR

[davidwei]

The Theorem 2 is so conter-intuitive. Even though I understand that it works for the concatenation, I still cannot believe it. Just for the given counterexample, does it mean that I have to dig a hole at every point when xsin(pi/x) touches 0? Then it fails to define on an interval close to a. Is it a counterexample for theorem 2? Also, how useful is theorem 2 and how can we actually use theorem 2 in solving problems? I found it is just deliberately designed to fix the problem but serve no other purpose...

[mond]

For the challenge question, I finally found this function (after watching like 5 times):
(Spoiler)
.
.
.
.
.
f(x) = 1 if x = 0
f(x) = 0 otherwise.
Is it correct?
I'm pretty sure it is.

[yukiorita]

I still don't clearly understand what 5:08 ~ 5:22 means. Does it mean that y = f(x)? In the next part, false theorem is showed as an incorrect example since first "then" part doesn't match the second "if" part, but I'm not sure how they are correctly matching in Theorem 1. I think I'm confused about the y part.
It would be really helpful if someone could explain it, thank you

[gregmiller]

Very good video. Nicely explained sir!!!

[ll-bcgn]

5:15 i do not understand why we consider the case when x is close to a when we are defining the meaning of the continuity of f at a?