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?

Rui: Why?

Vipul: What is the goal of this epsilon? Where
will it finally appear?

It will finally appear here. Is this captured?

Rui: Yes.

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?

Rui: Yes.

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.