People have thought, people have wondered 
about infinity for thousands of years. 
We're going to continue that tradition of 
thinking about infinity, but we're going 
to focus in on one specific instance of 
infinity in mathematics. 
What does it mean to say that the limit 
of a function equals infinity? 
Here's what it means. 
The limit of F of X, as X approaches A 
equals infinity means that F of X is as 
large as you like. 
Provided x is close enough to A. 
As a bit of a warning, I'm not saying 
that this number limit F of X, X 
approaches A is equal to this number, 
infinity is not a number. 
What I am doing here is attaching a 
precise meaning to the entire statement 
that the limit of F of X is infinity. 
So, that's what it means precisely, but 
what does it mean practically. 
Let's go to the board. 
This is an example, 
the limit of one over X squared as X 
approaches zero. 
Now this limit is not equal to a number. 
One over X squared is not getting close 
to a number when X is approaches zero. 
But one over X squared is as big as I 
want it to be if I'm willing to put X 
close to zero. 
For instance, if X is within a 1000th of 
zero, what happens? 
Well to say X is within a 1000th of zero 
is to say that X is between minus.00 
and.001. 
Now, X is also non zero here because I 
don't want to divide by zero. 
So I've got a non zero number within a 
1000th of zero. 
What does that mean about X squared? 
[SOUND]. 
Well that means that X squared is, it's 
bigger than zero because if I square any 
non zero number, it's positive and it's 
less than, [SOUND] a millionth. 
So if I take a number within a thousandth 
of zero and I square it, that was within 
a millionth of zero. 
And [SOUND] what does that mean about one 
over X squared? 
Well if X squared is within a millionth 
of zero, one over X squared is now 
bigger, [SOUND] than a million. 
What happened, right? 
I made one over x squared very large by 
making x close enough but not equal to 
zero. 
And there's nothing particularly special 
about these numbers. 
If I wanted one over x squared to be 
bigger than a trillion, I would just need 
x to be within a millionth of zero. 
This example is at once maybe too 
complicated because I've got some 
explicit numbers in here. 
And maybe this formula is not complicated 
enough to, to really get a sense of 
what's going on. 
Let's look at a slightly more complicated 
function. 
So what's the limit of X plus two over X 
squared minus 2x plus one as X approaches 
one. 
And this is a limit of a quotient. 
So your first temptation is to think the 
limit of a quotient the quotients the 
limits. 
But what's the limit of the denominator? 
The denominator is a polynomial. 
Polynomials are continuous. 
So I can evaluate the limit of the 
denominator just by plugging in one. 
And if I plug in one, I get one minus two 
plus one. 
That's zero. 
The limit of the denominator is zero. 
So I can't simply say that the limit of 
the quotient is the quotient of the 
limits in this case. 
Because the limit of the denominator is 
zero. 
And that limit law does not apply in this 
case. 
What am I going to do? 
Well I could also notice the denominator 
factors. 
X squared minus 2X plus one, I can 
rewrite that. 
[SOUND] X squared minus 2X plus one is X 
minus one squared. 
[SOUND] So instead of evaluating this 
limit, I could try to look at this limit, 
the limit of X plus two over X minus one 
squared as X approaches one. 
Now what do I know? 
The numerator is X plus two and if X is 
close to one, I can make X plus two close 
to three. 
What about the denominator? 
A number close to one, minus one squared, 
is a number close to zero. 
But not just a number close to zero. 
It's a positive number close to zero. 
So let me write this down. 
Alright. 
The numerator [SOUND] is about three. 
[SOUND] And the denominator [SOUND] is 
about what? 
Well, it's some small but positive 
number. 
[SOUND]. 
And what happens if I take a number, 
like, near three, and divide it by a 
number which is small and positive? 
That number can be as big as I like. 
So I can make this quantity as large as I 
like, if I make x close enough to one. 
Because I can make the numerator close to 
three, and I can make the denominator as 
close to zero and positive as I like. 
And if I take a number close to three and 
divide it by a number close to zero, I 
can make a number as large as I want. 
So this limit [SOUND] is equal to 
infinity. 
The homework includes even trickier 
situations. 
For instance, limit problems where x 
approaches negative infinity, instead of 
infinity. 
If you get stuck I encourage you to 
contact us. 
We're here to help you.