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# Introduction to limits (old)

An older video of Sal introducing the notion of a limit. Created by Sal Khan.

Video transcript

Welcome to the
presentation on limits. Let's get started with some--
well, first an explanation before I do any problems. So let's say I had-- let me
make sure I have the right color and my pen works. OK, let's say I had the limit,
and I'll explain what a limit is in a second. But the way you write it is you
say the limit-- oh, my color is on the wrong-- OK, let me
use the pen and yellow. OK, the limit as x
approaches 2 of x squared. Now, all this is saying is what
value does the expression x squared approach as
x approaches 2? Well, this is pretty easy. If we look at-- let me
at least draw a graph. I'll stay in this yellow color. So let me draw. x squared looks something
like-- let me use a different color. x square looks something
like this, right? And when x is equal to 2, y,
or the expression-- because we don't say what
this is equal to. It's just the expression-- x
squared is equal to 4, right? So a limit is saying, as x
approaches 2, as x approaches 2 from both sides, from numbers
left than 2 and from numbers right than 2, what does
the expression approach? And you might, I think, already
see where this is going and be wondering why we're even going
to the trouble of learning this new concept because it seems
pretty obvious, but as x-- as we get to x closer and closer
to 2 from this direction, and as we get to x closer
and closer to 2 to this direction, what does
this expression equal? Well, it essentially
equals 4, right? The expression is equal to 4. The way I think about it is as
you move on the curve closer and closer to the expression's
value, what does the expression equal? In this case, it equals 4. You're probably saying, Sal,
this seems like a useless concept because I could have
just stuck 2 in there, and I know that if this is-- say this
is f of x, that if f of x is equal to x squared, that f of 2
is equal to 4, and that would have been a no-brainer. Well, let me maybe give you one
wrinkle on that, and hopefully now you'll start to see what
the use of a limit is. Let me to define-- let me say
f of x is equal to x squared when, if x does not equal 2,
and let's say it equals 3 when x equals 2. Interesting. So it's a slight variation on
this expression right here. So this is our new f of x. So let me ask you a question. What is-- my pen still works--
what is the limit-- I used cursive this time-- what is the
limit as x-- that's an x-- as x approaches 2 of f of x? That's an x. It says x approaches 2. It's just like that. I just-- I don't know. For some reason, my brain
is working functionally. OK, so let me graph this now. So that's an equally
neat-looking graph as the one I just drew. Let me draw. So now it's almost the same as
this curve, except something interesting happens
at x equals 2. So it's just like this. It's like an x squared
curve like that. But at x equals 2 and
f of x equals 4, we draw a little hole. We draw a hole because it's
not defined at x equals 2. This is x equals 2. This is 2. This is 4. This is the f of x
axis, of course. And when x is equal to
2-- let's say this is 3. When x is equal to 2,
f of x is equal to 3. This is actually
right below this. I should-- it doesn't look
completely right below it, but I think you got
to get the picture. See, this graph is x squared. It's exactly x squared until
we get to x equals 2. At x equals 2, We have a
grap-- No, not a grap. We have a gap in the
graph, which maybe could be called a grap. We have a gap in the graph, and
then we keep-- and then after x equals 2, we keep moving on. And that gap, and that gap
is defined right here, what happens when x equals 2? Well, then f of x
is equal to 3. So this graph kind of goes--
it's just like x squared, but instead of f of 2 being 4, f
of 2 drops down to 3, but then we keep on going. So going back to the limit
problem, what is the limit as x approaches 2? Now, well, let's think
about the same thing. We're going to go-- this
is how I visualize it. I go along the curve. Let me pick a different color. So as x approaches 2 from this
side, from the left-hand side or from numbers less than 2, f
of x is approaching values approaching 4, right? f of x
is approaching 4 as x approaches 2, right? I think you see that. If you just follow along the
curve, as you approach f of 2, you get closer and closer to 4. Similarly, as you go from the
right-hand side-- make sure my thing's still working. As you go from the right-hand
side, you go along the curve, and f of x is also
slowly approaching 4. So, as you can see, as we
go closer and closer and closer to x equals 2, f of
whatever number that is approaches 4, right? So, in this case, the
limit as x approaches 2 is also equal to 4. Well, this is interesting
because, in this case, the limit as x approaches 2 of f
of x does not equal f of 2. Now, normally, this
would be on this line. In this case, the limit as you
approach the expression is equal to evaluating the
expression of that value. In this case, the limit isn't. I think now you're starting to
see why the limit is a slightly different concept than just
evaluating the function at that point because you have
functions where, for whatever reason at a certain point,
either the function might not be defined or the function kind
of jumps up or down, but as you approach that point, you still
approach a value different than the function at that point. Now, that's my introduction. I think this will give you
intuition for what a limit is. In another presentation, I'll
give you the more formal mathematical, you know,
the delta-epsilon definition of a limit. And actually, in the very next
module, I'm now going to do a bunch of problems
involving the limit. I think as you do more and more
problems, you'll get more and more of an intuition as
to what a limit is. And then as we go into drill
derivatives and integrals, you'll actually understand why
people probably even invented limits to begin with. We'll see you in the
next presentation.