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.