[MUSIC]. 
A lot of calculus is about understanding 
the qualitative features of functions. 
Nobody really cares about f of 17. 
But you might care about f of a big 
number, qualitatively. 
We're going to try to make this concept 
of big number a bit more precise. 
Well, here's how we're going to get 
around talking about a big number. 
We're going to talk about the limit. 
The limit of f of x as x approaches 
infinity equals to l. 
Means that f of x is as close as you want 
it to be to l, provided x is large 
enough. 
So instead of talking about just plugging 
in a big number, I'm going to say F at 
some big number, so to speak, is equal to 
L if I can make the output of F close to 
L by plugging in big enough numbers. 
At this point we've seen a bunch of 
different definitions of limits. 
But they're all united by a common theme. 
What if someone had asked us to cook up a 
definition of the limit of F of X equals 
infinity as X goes to infinity? 
Could we come up with a definition for 
this? 
Yeah, absolutely we can, here we go. 
The limit of F of X as X approaches 
infinity equals infinity means that I can 
make F of X as large as you want it to be 
provided X is large enough. 
Consistently when we're talking about 
infinity in limits we're never actually 
talking about a specific value. 
We're just talking about a value which is 
as big as you want it to be. 
Let's go do an example at the blackboard. 
There is a question. 
With the limit, of 2x over x+1 as x 
approaches infinity. 
Before we dive into this analytically, 
lets get some numeric evidence. 
This is my function, again fx)= of 2x 
over x+1.1. 
I want to know qualitatively, what 
happens when I plug in big numbers in 
this function? 
So, lets say f100). 
of 100. 
Well, that's not too hard to figure out. 
22 times 100 is 200 and 100 plus one is 
101. 
Now, 200 divided by 101 is pretty close 
to 2. 
And there's nothing too special about 
100, right? 
If I'd done this four million I would 
have gotten two million over a million in 
one, which would be even closer to 2. 
Numerically, it looks like this limit's 
two, but we just figured that out by 
plugging in some big numbers. 
I want a more rigorous argument, some 
analytic that is limit is actually equal 
to two. 
How I'm going to proceed? 
My first guess would be to use the limit 
law for quotients. 
This is the limit of a quotient which is 
the quotient of limits provided the 
limits exist and the limit of the 
denominator is none zero. 
Bad news here is the limits don't exist. 
The limit of the numerator is infinity 
which isn't a number. 
So I can't use my limit law for quotients 
here. 
Instead I'm going to sneak up on this 
limit problem by wearing a disguise. 
I'm going to multiply by a disguised 
version of one. 
I'm going to multiply by one over x 
divided by one over x. 
Now this is just one. 
Admittedly, I have changed the function. 
The function's not defined anymore at 
zero. 
But for large values of X, this doesn't 
affect anything. 
And I'm taking the limit as X goes to 
infinity. 
So I only care about agreement at large 
values of X. 
I'll do some algebra. 
This limit has now the limit of 2X times 
one over X. 
Which is two divided by the limit of X 
plus one times one over X. 
Which is one plus one over X. 
Maybe it doesn't look like I've made a 
lot of progress here, but this is a huge 
progress. 
The limit of the numerator is now a 
number, it's two. 
And the limit of the denominator is also 
a number. 
And a non zero number, at that. 
So I can use my limit law for quotients. 
This is the limit of a quotient, which is 
the quotient of a limit. 
It's the limit of two, the numerator 
divided by the limit of the denominator, 
as x approaches infinity. 
The limit of the numerator is just the 
limit of a constant, which is 2. 
The limit of the denominator is a limit 
of a sum, which is the sum of the limits 
provided the limits exist, and they do. 
The limit of 1 is just 1. 
And what's the limit of 1 over x as x 
approaches infinity? 
Well that's asking, what is 1 over x 
close to when x is very large? 
Well I can make 1 over x as close to zero 
as I like if I'm willing to make x large 
enough. 
So the limit of 1 over x as x approaches 
infinity is zero. 
That means my original limit is 2 over 1 
plus zero, which is 2. 
[MUSIC].