Let's do some more limit examples.
So let's get another problem.
If I had the limit as x approaches 3 of, let's say,
x squared minus 6x plus 9 over x squared minus 9.
So the first thing I like to do whenever I see any of these
limits problems is just substitute the number in and
see if I get something that makes sense, and
then we'd be done.
Well, usually we'd be done.
I don't want to make these sweeping statements.
If the function is continuous, we'd be done.
But if we put the 3 in the numerator, we get 3 squared,
which is 9, minus 18 plus 9.
So that equals 0.
And the denominator also-- let's see, 3 squared minus
9, that also equals 0.
So we don't like having 0/0.
My pen tool is malfunctioning again.
So we don't like getting 0, 0, 0, so is there any way we can
simplify this expression to maybe get it to an expression
that, when we evaluate it at x equals 3, we actually get
something that makes sense?
Well, whenever I see two of these polynomials here, and
they look, just by inspecting them, relatively easy to
factor, I like to factor them out because maybe there's the
same factor in the numerator and the denominator, and
then we can simplify it.
So let's say that this is the same thing as-- that looks
like it's x plus 3-- no, no, no, x minus 3.
This is x minus 3.
It actually looks like it's x minus 3 squared, but we're
just going to write x minus 3 times x minus 3, which is, of
course, x minus 3 squared.
And then in the denominator, you know how to factor these,
this is x plus 3 times x minus 3, all right?
So the limit as x approaches 3 of this expression is the same
thing as the limit as x approaches 3 of
this expression.
And, of course, there's nothing we can do to change the fact
that this function, or this expression, is undefined
at x equals 3.
But if we can simplify it, we can figure out
what it approaches.
Well, if we assume that x is any number but 3, we can cross
out these two terms because then they wouldn't be 0, right?
It only is 0 when x is equal to 3 because-- so in the numerator
and the denominator, we can cross this out.
And we can say-- and I'm not being very rigorous here, but
this is kind of how it's taught, and I think you get the
intuition-- that this is the same thing as the limit as x
approaches 3 of x minus 3 over x plus 3.
Now let's just try to stick the x in and see what we get.
Well, in the numerator, we get 3 minus 3.
We still get 0.
But in the denominator here, we get 6, right?
3 plus 3 is 6.
So now we get a good number.
0 or 6, well, that's a real number, so it's 0.
0/6 is 0.
So that was interesting.
The first time we did it, we got the answer 0/0.
And now we get the answer 0 by simplifying.
But, of course, it's very important to remember that
this expression is not defined at x equals 3.
It's defined everywhere but, but if we were to graph it, and
I encourage you to do so, you would see that as you get
closer and closer to x equals 3, the value of this
expression will equal 0.
And I know what you're thinking.
Well, this was 0/0.
Is every time I get 0/0 going to end up just becoming 0 when
I evaluate the expression?
Well, let's explore that.
Let me clear this.
Let's say what is-- pen is not working-- the limit as x
approaches 1 of x squared minus x minus 2.
No, let's say x squared plus x minus 2.
As you can see, I do all this in my head, and
I'm prone to mistakes.
And all of that over x minus 1.
Well, once again, if we just evaluate it, let's see what
happens when x equals 1.
You get 1 squared plus 1, so it's 2 minus 2.
You get 0/0.
So once again, we get 0/0, and we have to do something to
this maybe to simplify it.
Well, let's factor the top.
So that's the same thing as the limit as x approaches 1.
Well, that's x minus 1 times x plus 2, right?
And I think you'll often discover when you see a lot of
limit problems that even if this top factor, if this top
expression, is hard to factor, chances are, one of the things
in the denominator that are making this expression
undefined is probably a factor up here.
So sometimes you might get a more complex thing that isn't
as easy to factor as this, but a good starting point is to
guess that one of the factors is going to be in the bottom
expression because that's kind of the trick of these problems,
to just simplify the expression.
So once again, if we assume that x does not equal 1, and
this expression would not be 0 and this would not be 0,
then these two could be canceled out.
And we get that this is just the same thing as the limit as
x approaches 1 of x plus 2.
Well, now this is pretty easy.
What's the limit as x approaches 1 of x plus 2?
Well, you just stick 1 in there, and you get 3.
So it's interesting.
When we just tried to evaluate the expression at
x equals 1, we got 0/0.
And in the previous example, we saw that it evaluated out when
you simplified it to 0, and in this example, it came out to 3.
And I really encourage you, if you have a graphing calculator,
graph these functions that we're doing and see and show
yourself visually that it's true, that the limit as you
approach, say, x equals 1 actually does approach the
limits that were solving for.
And make up your own problems.
Hell, that's what I'm doing.
So you could prove it to yourself.
So let's do another.
Let's do one that I think is pretty interesting.
Let's say what's the limit as x approaches infinity?
The limit as x approaches infinity of, let's say, x
squared plus 3 over x to the third.
So the way I think about these problems as they approach
infinity, just think about what happens when you get
really, really, really large values of x.
And kind of a cheating way of doing this is, if you have a
calculator, even if you don't have a calculator, put
in huge numbers here.
See what happens when x is a million, see what happens when
x is a billion, see what happens when x is a trillion,
and I think you'll get the point.
You'll see what-- if there is a limit here, you'll
see what it's going to.
But the way I think about it is, in the numerator, kind of
the fastest-growing term here is the x squared term, right?
This is the fastest-growing term here.
In the denominator, what's the fastest-growing term?
Well, in the denominator, the fastest-growing term
is this x to the third.
Well, what's going to grow faster, x to the
third or x squared?
Well, yeah, x to the third's going to grow a lot
faster than x squared.
So this denominator, as you get larger and larger and larger
values of x, is going to grow a lot faster than that numerator.
So you could imagine if the denominator's growing much,
much, much faster than the numerator, as you get larger
and larger numbers, you're going to get a smaller and
smaller and smaller fraction, right?
It's going to approach 0.
And so as you go to infinity, it approaches 0.
I know that I kind of just hand waved, but that's really
how you think about it.
Another way you could do it is you could actually
divide this fraction.
You could actually divide this rational expression, and you'll
get something like 1/x plus something, something,
something, and then you'd also see, oh, well, the limit as x
approaches infinity of 1/x is also 0.
Let's do one more.
I'll do this fast so I can confuse you.
The limit as x approaches infinity of 3x squared plus
x over 4x squared minus 5.
These problems kind of look confusing sometimes, but
they're really easy.
You just have to think about what happens as you get
really large values of x.
Well, as you get really large values of x, these small terms,
these ones that don't grow as fast as these large terms,
kind of don't matter anymore, right, because you're getting
really large values of x.
And this case, these don't matter anymore, and then
these two x terms grow at the same pace, right?
And they'll always be kind of growing in
this ratio of 3 to 4.
So the limit here is actually that easy.
It's 3/4.
So what you do is you just figure out what's the
fastest-growing term on the top, what's the fastest-growing
term on the bottom, and then figure out what it approaches.
If they're the same term, then they kind of cancel out, and
you say the limit approaches 3/4.
It's a very nonrigorous way of doing it, but it gets
you the right answer.
See you in the next presentation.