Vipul: Ok, so in this talk I'm going to
do the conceptual definition
of limit, which is important for a number of reasons. The main reason
is it allows you to construct definitions of limit, not just for this
one variable, function of one variable, two sided limit which you have
hopefully seen before you saw this video. Also for a number of other
limit cases which will include limits to infinity, functions of two
variables, etc. So this is a general blueprint for thinking about
limits. So let me put this definition here in front for this. As I am
going, I will write things in more general. So the starting thing is...
first of all f should be defined around the
point c, need not be
defined at c, but should be defined everywhere around c. I won't write
that down, I don't want to complicate things too much. So we start
with saying for every epsilon greater than zero. Why are we picking
this epsilon greater than zero?
Vipul: What is the goal of this epsilon? Where will it finally appear?
It will finally appear here. Is this captured?
Vipul: Which means what we actually are picking
when we...if you've
seen the limit as a game video or you know how to make a limit as a
game. This first thing has been chosen by the skeptic, right, and the
skeptic is trying to challenge the prover into trapping f(x) within L - epsilon to
L + epsilon. Even if you haven't seen that [the game], the main focus of
picking epsilon is to pick this interval surrounding L. So instead of
saying, for every epsilon greater than zero, let's say for every
choice of neighborhood of L. So what I mean by that, I have not
clearly defined it so this is a definition which is not really a
definition, sort of the blueprint for definitions. It is what you fill
in the details [of] and get a correct definition. So by neighborhood,
I mean, in this case, I would mean something like (L - epsilon, L +
epsilon). It is an open interval surrounding L. Ok, this one. The
conceptual definition starts for every choice of neighborhood of
L. The domain neighborhood, I haven't really defined, but that is the
point, it is the general conceptual definition. There exists...what
should come next? [ANSWER!]
Rui: A delta? Vipul: That is what the concrete definition
says, but what would the conceptual thing say?
Rui: A neighborhood. Vipul: Of what? [ANSWER!]
Rui: Of c. Vipul: Of c, of the domain. The goal of picking
delta is to find a neighborhood of c. Points to the immediate
left and immediate right of c. There exists a choice of neighborhood
of c such that, by the way I sometimes abbreviate, such that,
as s.t., okay, don't get confused by that. Okay, what next? Let's
bring out the thing. The next thing is for all x with |x - c| less than
... all x in the neighborhood except the point c itself. So what should
come here? For all x in the neighborhood of c, I put x not equal to c.
Is that clear?
Vipul: x not equal to c in the neighborhood
chosen for c. The reason
we're excluding the point c that we take the limit at the point and we
just care about stuff around, we don't care about what is happening at
the point. For c...this chosen neighborhood...I am writing the black
for choices that the skeptic makes and the red for the choices the
prover makes, actually that's reverse of what I did in the other
video, but that's ok. They can change colors. If you have seen that
limit game thing, this color pattern just [means] ... the black
matches with the skeptic choices and the red matches what the prover
chooses. If you haven't seen that, it is not an issue. Just imagine
it's a single color.
What happens next? What do we need to check
in order to say this limit
is L? So f(x) should be where?
Rui: In the neighborhood of L.
Vipul: Yeah. In the concrete definition we
said f(x) minus L is less
than epsilon. Right, but that is just stating that f(x) is in the
chosen neighborhood. So f(x) is in the chosen neighborhood of...Now
that we have this blueprint for the definition. This is a blueprint
for the definition. We'll write it in blue. What I mean is, now if I
ask you to define a limit, in a slightly different context; you just
have to figure out in order to make this rigorous definition. What
word do you need to understand the meaning of? [ANSWER!]
Rui: Neighborhood. Vipul: Neighborhood, right. That's the magic
word behind which I am hiding the details. If you can understand
what I mean by neighborhood then you can turn this into a concrete definition.