[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.