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Vipul: In this video, I'm going to go over the usual definition of limit and think of
it in terms of a game.
The game is as follows.
Consider this statement.
You are saying limit as x approaches c of f(x) is L.
Okay.
There are two players to this game.
One is the prover and one is the skeptic.
The prover's goal is to show that this claim is true so the prover is trying to convince
the skeptic that this limit as x approaches c of f(x) is L,
the skeptic will try to ask tough questions and see if the prover can still manage to show this.
The way the game is structured is as follows.
Let me just go over the individual components of the statement for the limit and I will
translate each one.
I will explain the game and then explain how it corresponds to the definition you've seen.
We begin with the skeptic chooses epsilon > 0.
This is the part of the definition which reads for every epsilon > 0.
That's the first clause of the definition and that's basically the skeptic is choosing
epsilon > 0.
What is the skeptic trying to do when choosing epsilon > 0?
What the skeptic is effectively doing is choosing this interval L -- epsilon to L + epsilon.
The skeptic is effectively trying to choose this interval L -- epsilon to L + epsilon.
What is the skeptic trying the challenge the prover into doing when picking this interval? [ANSWER!]
Rui: Whether the prover can trap.
Vipul: The skeptic is trying to challenge (and this will become a clearer a little later).
The idea is, the skeptic is trying to challenge the prover into trapping the function when
the input x is close to c, trapping the function output within this interval and that's
not clear which is why we need to continue its definition.
The prover chooses. What does the prover choose? [ANSWER!]
Rui: delta.
Vipul: delta > 0 and this corresponds to the next part of the definition which says
there exists delta > 0.
In this picture, which I have up here, this is the value c.
This is c + delta and this is c -- delta.
This is c and L, so c is the x coordinate, L is the function value or limited the function value.
The skeptic chooses this strip like this from L -- epsilon to L + epsilon by choosing epsilon
so the skeptic just chooses the number absent what it is effectively doing is to choose
this strip, L -- epsilon to L + epsilon. The prover then chooses a delta.
What's the prover effectively choosing?
The prover is effectively choosing this interval.
Okay so that's this interval.
It is c -- delta to c + delta except you don't really care about the point c itself,
(but that's a little subtlety we don't have to bother about), so the skeptic is choosing
the interval like this. The prover is choosing the interval like this.
How is the skeptic choosing the interval? By just specifying the value of epsilon.
How is the prover choosing [the interval around c]?
By just specifying a value of delta. Okay. Now what does the skeptic now do? [ANSWER!]
Rui: Skeptic will check.
Vipul: There is something more to choose (right?) before checking.
What does the definition say? For every epsilon > 0 there exists a delta greater than zero
such that ... [COMPLETE!]
Rui: For every.
Vipul: For every x such that something. The skeptic can now pick x.
Rui: That's what I meant by checking.
Vipul: The skeptic could still, like, pick a value to challenge the prover.
The skeptic chooses x but what x can the skeptic choose?
Rui: Within the...
Vipul: This interval which the prover has specified.
The skeptic is constrained to choose x within the interval.
That's the same as c -- delta ... Is this all coming?
Rui: Yes.
Vipul: c -- delta, c union c to c + delta.
The way it's written is for every x in this interval.
Lot of people write this in a slightly different way.
They write it as ...
(You should see the definition video before this.)
(I'm sort of assuming that you have seen the definition -- this part [of the screen] so you can map it)
so a lot of people write it like this.
It is just saying x is within delta distance of c but it's not equal to c itself.
Now it's time for the judge to come in and decide who has won.
How does the judge decide? [ANSWER!]
Rui: For the x that the skeptic chooses and see the corresponding y.
Vipul: The f(x) value.
Rui: If the f(x) value is within the horizontal strip then the prover wins.
Vipul: If |f(x) -- L| < epsilon which is the same as saying f(x) is in what interval? [ANSWER!]
L- epsilon to L + epsilon then the prover wins. Otherwise? [ANSWER!]
Rui: The skeptic wins.
[But] the skeptic can choose a really dumb [stupid] x.
Vipul: That's actually the next question I want to ask you.
What does it actually mean to say that this statement is true?
Is it just enough that the prover wins? That's not enough.
What do you want to say to say that this statement is true?
Rui: For every x in the interval.
Vipul: For every x but not only for every x you should also say for every epsilon.
All the moves that the skeptic makes, the prover should have a strategy, which works for all of them.
So, this statement is true [if] ...
This is true if the prover has what for the game? [ANSWER!]
Rui: Winning strategy. Vipul: Winning what? Rui: Strategy.
Vipul: Yeah. True if the prover has a winning strategy.
It is not just enough to say that the prover won the game some day but the prover should
be able to win the game regardless of how smart the skeptic is or regardless of how
experienced the skeptic is or regardless of how the skeptic plays.
That's why all the moves of the skeptic are prefaced with a "for every." Right?
Whereas all the moves of the prover are prefaced, (well there is only one move really of the
prover) are prefaced with "there exists" because the prover controls his own choices.
When it is the prover's turn it's enough to say "there exists" but since the prover doesn't
control what the skeptic does all the skeptic moves are prefaced with "for every."
By the way, there is a mathematical notation for these things.
There are mathematical symbols for these, which I'm not introducing in this video,
but if you have seen them and got confused then you can look at the future video where
I explain the mathematical symbols.