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# Formal definition of limits Part 2: building the idea

Some background intuition to make the formal definition of a limit make intuitive sense. Created by Sal Khan.

Video transcript

Let's try to come up with
a mathematically rigorous definition for what
this statement means. The statement that
the limit of f of x as x approaches c
is equal to L. So let's say that this means that
you can get f of x as close to L as you want. I'll put that in
quotes right over here, because it's kind
of a little loosey goosey as how close is that. But as close as
you want by getting x sufficiently close to c. So another way of saying
this is, if you tell me, hey, I want to get my f of x to
be within 0.5 of this limit. Then you're telling me if
this limit is actually true, you should be able to
hand me a value around c. That if x is within
that range, then f of x is definitely going to be
as close to L as I desire. So let me draw that out to
make it a little bit clearer. And I'm going to zoom in. I'm going to draw
another diagram. So let's say that this right
over here is my y-axis. And I'm going to zoom in. I'm going to draw a slightly
different function, just so we can really focus on what's
going on around here. The range is around c, and the
range is around L. So that's x. This right over here is y. Let's say that this is c. And let's just zoom
in on our function. So let's say our function
looks, is doing something like, let's say it does something
like, let's see, I don't want it to
be defined at c. At least just for
the-- it could be. You can always find a limit
even where is defined. But let's say our function
looks something like that. And it can have a little kink
in it, the way I drew it. So it looks something like this. It's undefined. Let me draw it a
little bit different. So it is undefined
when x is equal to c. So this is the point
where there's a hole. It is undefined when
x is equal to c. So it even has a little
kink in it, just like that. And what we want to do
is prove that the limit, as x, the limit of f of x--
and let me make it clear, this is the graph
of y is equal to f of x-- we want to get an idea
for what this definition is saying. If we're claiming that the limit
of f of x, as x approaches c, is L. So conceptually, we
get the gist already. We already get the gist that
this right over here is L. But what is this
definition saying? Well, it's saying that you
can get f of x as close to L as you want. So if you tell
someone, I want to get f of x within a
certain range of L, then if this limit
is actually true, if the limit of f of
x as x approaches c really is equal to
L, then they should be able to find
a range around c. That as long as x is
around that range, your f of x is going to be
in the range that you want. So let me actually go
through that exercise. It really is a little
bit like a game. So someone comes up to
you and says, well, OK. I don't necessarily
believe that you're claiming the limit of f of x as
x approaches c is equal to L. I'm not really sure
if that's the case. But I agree with
this definition. So I want to get within 0.5. I want to get f of x within 0.5
of L. So this right over here would be L plus 0.5. And this right over
here is L minus 0.5. And then you say, fine. I'm going to give
you a range around c, that if you take any x within
that range, your f of x is always going to fall in
this range that you care about. And so you look at this-- and
obviously we haven't explicitly defined this function. But you can even eyeball it, the
way this function is defined. It won't be that easy
for all functions. But you look at it like this. And you say that this
value, just the way it's drawn right
over here, let's say that this is c minus 0.25. And let's say that this value
right over here is c plus 0.25. And so you tell
them, look, as long as you get x within
0.25 of c, so as long as your x's are sitting
someplace over here, the corresponding f of x is
going to sit in the range that you care about. And you say, OK, fine. You won that round. Let me make it even tighter. Maybe instead of saying within
the 0.5, I want to get within is 0.05. And then you'd have to do
this exercise again and find another range. And in order for
this to be true, you would have to be able
to do this for any range that they give you. For any range around
L that they give you, you have to be able to get
f of x within that range by finding a range around c. That as long as x is
that range around c, f of x is going to
sit within that range. So I'll let you think
about that a little bit. There's a lot to think about. But hopefully this made sense. We did it for the particular
example of someone hands you the 0.5, I want
f of x within the 0.5 of L, and you say, well, as long
as x is within 0.25 of c, you're going to match it. You need to be able to
do that for any range they give you around
L. And then this limit will definitely be true. So in the next video, we
will now generalize that. And that will really bring us
to the famous epsilon delta definition of limits.