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## Formal definition of limits (epsilon-delta)

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# Formal definition of limits Part 1: intuition review

## Video transcript

Let's review our intuition
of what a limit even is. So let me draw some axes here. So let's say this is my y-axis,
so try to draw a vertical line. So that right over
there is my y-axis. And then let's say
this is my x-axis. I'll focus on the
first quadrant, although I don't have to. So let's say this right
over here is my x-axis. And then let me draw a function. So let's say my function
looks something like that, could look like anything,
but that seems suitable. So this is the function
y is equal to f of x. And just for the sake of
conceptual understanding, I'm going to say it's
not defined at a point. I didn't have to do this. You can find the limit as
x approaches a point where the function
actually is defined, but it becomes that much more
interesting, at least for me, or you start to understand
why a limit might be relevant where a function is not
defined at some point. So the way I've drawn
it, this function is not defined when
x is equal to c. Now, the way that we've
thought about a limit is what does f of x
approach as x approaches c? So let's think about
that a little bit. When x is a reasonable
bit lower than c, f of x, for our function that we
just drew, is right over here. That's what f of x is going to
be equal. y is equal to f of x. When x gets a little bit
closer, then our f of x is right over there. When x gets even closer,
maybe really almost at c, but not quite at c, then
our f of x is right over here. And the way we see it,
we see that our f of x seems to be-- as x gets
closer and closer to c it looks like our f
of x is getting closer and closer to some
value right over there. Maybe I'll draw it
with a more solid line. And that was actually
only the case when x was getting closer
to c from the left, from values of x less than c. But what happens as we
get closer and closer to c from values of x
that are larger than c? Well, when x is over here,
f of x is right over here. And so that's what f of x
is, is right over there. When x gets a little bit
closer to c, our f of x is right over there. When x is just very slightly
larger than c, then our f of x is right over there. And you see, once
again, it seems to be approaching
that same value. And we call that value,
that value that f of x seems to be approaching
as x approaches c, we call that value
L, or the limit. And so the way we
would denote it is we would call that the limit. We don't have to call
it L all the time, but it is referred
to as the limit. And the way that
we would kind of write that mathematically is
we would say the limit of f of x as x approaches
c is equal to L. And this is a fine conceptual
understanding of limits, and it really will
take you pretty far, and you're ready to
progress and start thinking about taking a lot of limits. But this isn't a very
mathematically-rigorous definition of limits. And so this sets us
up for the intuition. In the next few videos,
we will introduce a mathematically-rigorous
definition of limits that will
allow us to do things like prove that the
limit as x approaches c truly is equal to L.