[MUSIC] So now we learn what the official definition of limit is.
To say that the limit of f(x) = L as x approaches a, means the following.
It means that for all epsilon bigger than zero, this backwards three is the real
number epsilon or the Greek letter, a variable. So for all epsilon greater than
zero, there's a delta greater than zero, this is the Greek letter delta.
So, for all epsilon greater than zero, there's a delta greater than zero.
So that if, this, if the absolute value of x - a is between zero and delta, then,
the absolute value of f(x) - L is less than epsilon.
When you say it like that, I think it's really hard to see how this has any
relationship to what a more intuitive description of this limit statement might
be. I mean, what's this trying to get at?
It's trying to say f(x) is as close as I want to L by making x sufficiently close
to a. So, how, how to reconcile those, those
two perspectives, right? How does this have anything to do with things being
close. The key, take a look at this absolute
value of the difference, right?
The absolute value of x - a is the distance between x and a.
So to say that the distance between x and a is between zero and delta is to say
that x is within delta of a, alright? The distance from x to a is less than delta.
And to say that the distance between x and a is bigger than zero is just to say
that the distance between x and, and a, you know, isn't zero, right, x isn't a.
So I can rewrite that, maybe in a little bit easier way.
[SOUND] So instead of saying that, it's the same thing to say, if x is not equal
to a, so the absolute value of x - a isn't zero, and x is within delta of a.
So the distance between x and a is delta. And I can do the same thing to this
absolute value of a difference, alright? The absolute value of f(x) - L, that's
the distance, between f(x) and L. And the sum of the distance between f(x)
and L is less than epsilon. Well, that just means that f(x) is within
epsilon of L. So, I'll rewrite that as that. [SOUND]
Here we go. [SOUND] Then f(x) is within epsilon of L.
So, I think when, when you write it like this, it makes a little bit more sense,
right? To say that the limit of f(x) = L as x
approaches a, means that for all numbers epsilon,
epsilon is measuring how close I want f(x) to L,
then, there's some corresponding number delta, which is how close x has to be to
a. So that whenever x is that close, delta,
within delta of a, then f(x) is really within epsilon of L.
And to say that the limit of f(x) = L means that no matter which epsilon I
choose, there's some corresponding delta, so that whenever x is within delta of a,
then f(x) is within epsilon of L. Now, how this actually gets played out
in, in more concrete situations can be, you know, kind of complicated, but this
is really the official definition of what it means to say that the limit of f(x)
equals L as x goes to a. [MUSIC] And we're going to be trying to unpack this
definition to see what it might mean in some specific cases.