Epsilon-delta definition of limits - Video Lecture - Engineering

In the last video,
we tried to come up
with a somewhat rigorous
definition of what a limit is,
where we say when you
say that the limit of f
of x as x approaches C is
equal L, you're really saying--
and this is the somewhat
rigorous definition--
that you can get f
of x as close as you
want to L by making x
sufficiently close to C.
So let's see if we can put
a little bit of meat on it.
So instead of saying
as close as you want,
let's call that some
positive number epsilon.
So I'm just going to use the
Greek letter epsilon right over
there.
So it really turns into a game.
So this is the game.
You tell me how close you
want f of x to be to L.
And you do this by giving me
a positive number that we call
epsilon, which is really
how close you want
f of x to be to L. So you give
a positive number epsilon.
And epsilon is how
close do you want to be?
How close?
So for example, if
epsilon is 0.01,
that says that you want f of x
to be within 0.01 of epsilon.
And so what I then
do is I say well, OK.
You've given me that epsilon.
I'm going to find you another
positive number which we'll
call delta-- the lowercase
delta, the Greek letter
delta-- such that where
if x is within delta of C,
then f of x will be within
epsilon of our limit.
So let's see if these are
really saying the same thing.
In this yellow definition
right over here,
we said you can
get f of x as close
as you want to L by making
x sufficiently close to C.
This second definition,
which I kind of
made as a little bit more of a
game, is doing the same thing.
Someone is saying how close
they want f of x to be to L
and the burden is then to
find a delta where as long
as x is within delta
of C, then f of x
will be within
epsilon of the limit.
So that is doing it.
It's saying look, if we are
constraining x in such a way
that if x is in that
range to C, then
f of x will be as
close as you want.
So let's make this
a little bit clearer
by diagramming right over here.
You show up and you
say well, I want
f of x to be within
epsilon of our limit.
This point right over here
is our limit plus epsilon.
And this right over here might
be our limit minus epsilon.
And you say, OK, sure.
I think I can get your f of x
within this range of our limit.
And I can do that by
defining a range around C.
And I could visually
look at this boundary.
But I could even go
narrower than that boundary.
I could go right over here.
Says OK, I meet your challenge.
I will find another
number delta.
So this right over
here is C plus delta.
This right over here
is C minus-- let
me write this down--
is C minus delta.
So I'll find you some
delta so that if you
take any x in the range
C minus delta to C
plus delta-- and maybe
the function's not even
defined at C, so
we think of ones
that maybe aren't C, but
are getting very close.
If you find any x in
that range, f of those
x's are going to be as close
as you want to your limit.
They're going to be within
the range L plus epsilon or L
minus epsilon.
So what's another
way of saying this?
Another way of saying this
is you give me an epsilon,
then I will find you a delta.
So let me write this in a
little bit more math notation.
So I'll write the
same exact statements
with a little bit
more math here.
But it's the exact same thing.
Let me write it this way.
Given an epsilon
greater than 0-- so
that's kind of the
first part of the game--
we can find a delta
greater than 0, such
that if x is within delta of C.
So what's another way of saying
that x is within delta of C?
Well, one way you
could say, well,
what's the distance
between x and C
is going to be less than delta.
This statement is true for any
x that's within delta of C.
The difference between the two
is going to be less than delta.
So that if you pick an x that
is in this range between C
minus delta and C
plus delta, and these
are the x's that satisfy
that right over here, then--
and I'll do this in
a new color-- then
the distance between your
f of x and your limit--
and this is just the
distance between the f of x
and the limit, it's going
to be less than epsilon.
So all this is saying is, if
the limit truly does exist,
it truly is L, is if you give
me any positive number epsilon,
it could be super, super small
one, we can find a delta.
So we can define
a range around C
so that if we take
any x value that
is within delta of C,
that's all this statement
is saying that the
distance between x and C
is less than delta.
So it's within delta
of C. So that's
these points right over here.
That f of those
x's, the function
evaluated at those x's is
going to be within the range
that you are specifying.
It's going to be within
epsilon of our limit.
The f of x, the difference
between f of x, and your limit
will be less than epsilon.
Your f of x is going to
sit some place over there.
So that's all the epsilon-delta
definition is telling us.
In the next video, we will
prove that a limit exists
by using this
definition of limits.