Let me draw a function that would be interesting
to take a limit of.
And I'll just draw it visually for now, and we'll do some
specific examples a little later.
So that's my y-axis, and that's my x-axis.
And let;s say the function looks something like--
I'll make it a fairly straightforward function
--let's say it's a line, for the most part.
Let's say it looks just like, accept it has a
hole at some point.
x is equal to a, so it's undefined there.
Let me black that point out so you can see that
it's not defined there.
And that point there is x is equal to a.
This is the x-axis, this is the y is equal f of x-axis.
Let's just say that's the y-axis.
And let's say that this is f of x, or this is
y is equal to f of x.
Now we've done a bunch of videos on limits.
I think you have an intuition on this.
If I were to say what is the limit as x approaches a,
and let's say that this point right here is l.
We know from our previous videos that-- well first of all
I could write it down --the limit as x approaches
a of f of x.
What this means intuitively is as we approach a from either
side, as we approach it from that side, what does
f of x approach?
So when x is here, f of x is here.
When x is here, f of x is there.
And we see that it's approaching this l right there.
And when we approach a from that side-- and we've done
limits where you approach from only the left or right side,
but to actually have a limit it has to approach the same thing
from the positive direction and the negative direction --but as
you go from there, if you pick this x, then this is f of x.
f of x is right there.
If x gets here then it goes here, and as we get closer and
closer to a, f of x approaches this point l, or this value l.
So we say that the limit of f of x ax x approaches
a is equal to l.
I think we have that intuition.
But this was not very, it's actually not rigorous at all
in terms of being specific in terms of what we
mean is a limit.
All I said so far is as we get closer, what does
f of x get closer to?
So in this video I'll attempt to explain to you a definition
of a limit that has a little bit more, or actually a lot
more, mathematical rigor than just saying you know, as x gets
closer to this value, what does f of x get closer to?
And the way I think about it's: kind of like a little game.
The definition is, this statement right here means that
I can always give you a range about this point-- and when I
talk about range I'm not talking about it in the whole
domain range aspect, I'm just talking about a range like you
know, I can give you a distance from a as long as I'm no
further than that, I can guarantee you that f of x is go
it not going to be any further than a given distance from l
--and the way I think about it is, it could be viewed
as a little game.
Let's say you say, OK Sal, I don't believe you.
I want to see you know, whether f of x can get within 0.5 of l.
So let's say you give me 0.5 and you say Sal, by this
definition you should always be able to give me a range
around a that will get f of x within 0.5 of l, right?
So the values of f of x are always going to be right in
this range, right there.
And as long as I'm in that range around a, as long as I'm
the range around you give me, f of x will always be at least
that close to our limit point.
Let me draw it a little bit bigger, just because I think
I'm just overriding the same diagram over and over again.
So let's say that this is f of x, this is the hole point.
There doesn't have to be a hole there; the limit could equal
actually a value of the function, but the limit is more
interesting when the function isn't defined there
but the limit is.
So this point right here-- that is, let me draw the axes again.
So that's x-axis, y-axis x, y, this is the limit point
l, this is the point a.
So the definition of the limit, and I'll go back to this in
second because now that it's bigger I want explain it again.
It says this means-- and this is the epsilon delta definition
of limits, and we'll touch on epsilon and delta in a second,
is I can guarantee you that f of x, you give me any
distance from l you want.
And actually let's call that epsilon.
And let's just hit on the definition right
from the get go.
So you say I want to be no more than epsilon away from l.
And epsilon can just be any number greater, any real
number, greater than 0.
So that would be, this distance right here is epsilon.
This distance there is epsilon.
And for any epsilon you give me, any real number-- so this
is, this would be l plus epsilon right here, this would
be l minus epsilon right here --the epsilon delta definition
of this says that no matter what epsilon one you give me, I
can always specify a distance around a.
And I'll call that delta.
I can always specify a distance around a.
So let's say this is delta less than a, and this
is delta more than a.
This is the letter delta.
Where as long as you pick an x that's within a plus delta and
a minus delta, as long as the x is within here, I can guarantee
you that the f of x, the corresponding f of x is going
to be within your range.
And if you think about it this makes sense right?
It's essentially saying, I can get you as close as you want to
this limit point just by-- and when I say as close as you
want, you define what you want by giving me an epsilon; on
it's a little bit of a game --and I can get you as close as
you want to that limit point by giving you a range around the
point that x is approaching.
And as long as you pick an x value that's within this range
around a, long as you pick an x value around there, I can
guarantee you that f of x will be within the range
you specify.
Just make this a little bit more concrete, let's say you
say, I want f of x to be within 0.5-- let's just you know, make
everything concrete numbers.
Let's say this is the number 2 and let's say this is number 1.
So we're saying that the limit as x approaches 1 of f of x-- I
haven't defined f of x, but it looks like a line with the hole
right there, is equal to 2.
This means that you can give me any number.
Let's say you want to try it out for a couple of examples.
Let's say you say I want f of x to be within point-- let me do
a different color --I want f of x to be within 0.5 of 2.
I want f of x to be between 2.5 and 1.5.
Then I could say, OK, as long as you pick an x within-- I
don't know, it could be arbitrarily close but as long
as you pick an x that's --let's say it works for this function
that's between, I don't know, 0.9 and 1.1.
So in this case the delta from our limit point is only 0.1.
As long as you pick an x that's within 0.1 of this point, or 1,
I can guarantee you that your f of x is going to
lie in that range.
So hopefully you get a little bit of a sense of that.
Let me define that with the actual epsilon delta, and this
is what you'll actually see in your mat textbook, and then
we'll do a couple of examples.
And just to be clear, that was just a specific example.
You gave me one epsilon and I gave you a delta that worked.
But by definition if this is true, or if someone writes
this, they're saying it doesn't just work for one specific
instance, it works for any number you give me.
You can say I want to be within one millionth of, you know, or
ten to the negative hundredth power of 2, you know, super
close to 2, and I can always give you a range around this
point where as long as you pick an x in that range, f of x will
always be within this range that you specify, within that
were you know, one trillionth of a unit away from
the limit point.
And of course, the one thing I can't guarantee is what
happens when x is equal to a.
I'm just saying as long as you pick an x that's within my
range but not on a, it'll work.
Your f of x will show up to be within the range you specify.
And just to make the math clear-- because I've been
speaking only in words so far --and this is what we see the
textbook: it says look, you give me any epsilon
greater than 0.
Anyway, this is a definition, right?
If someone writes this they mean that you can give them any
epsilon greater than 0, and then they'll give you a delta--
remember your epsilon is how close you want f of x to be
to your limit point, right?
It's a range around f of x --they'll give you a delta
which is a range around a, right?
Let me write this.
So limit as approaches a of f of x is equal to l.
So they'll give you a delta where as long as x is no more
than delta-- So the distance between x and a, so if we pick
an x here-- let me do another color --if we pick an x here,
the distance between that value and a, as long as one, that's
greater than 0 so that x doesn't show up on top of a,
because its function might be undefined at that point.
But as long as the distance between x and a is greater
than 0 and less than this x range that they gave you,
it's less than delta.
So as long as you take an x, you know if I were to zoom the
x-axis right here-- this is a and so this distance right here
would be delta, and this distance right here would be
delta --as long as you pick an x value that falls here-- so as
long as you pick that x value or this x value or this x value
--as long as you pick one of those x values, I can guarantee
you that the distance between your function and the limit
point, so the distance between you know, when you take one of
these x values and you evaluate f of x at that point, that the
distance between that f of x and the limit point is
going to be less than the number you gave them.
And if you think of, it seems very complicated, and I have
mixed feelings about where this is included in most
calculus curriculums.
It's included in like the, you know, the third week before you
even learn derivatives, and it's kind of this very mathy
and rigorous thing to think about, and you know, it tends
to derail a lot of students and a lot of people I don't think
get a lot of the intuition behind it, but it is
mathematically rigorous.
And I think it is very valuable once you study you know, more
advanced calculus or become a math major.
But with that said, this does make a lot of sense
intuitively, right?
Because before we were talking about, look you know, I can get
you as close as x approaches this value f of x is going
to approach this value.
And the way we mathematically define it is, you say Sal,
I want to be super close.
I want the distance to be f of x [UNINTELLIGIBLE].
And I want it to be 0.000000001, then I can always
give you a distance around x where this will be true.
And I'm all out of time in this video.
In the next video I'll do some examples where I prove the
limits, where I prove some limit statements using
this definition.
And hopefully you know, when we use some tangible numbers, this
definition will make a little bit more sense.
See you in the next video.