[MUSIC] Limits are probably the most 
important concept in this course. 
So we should really have a definition of 
what we mean by limit. 
Now here is what we mean by limits. 
To say that the limit of f of x as x 
approaches a is equal to L means that f 
of x can be as close to L as desired by 
making x close enough to a. 
There is a tons of subtlety to this 
definition so it's worth to look at an 
example. 
So let's take a look at this function. 
This is the function that takes an input 
x and spits out x^second minus one 
divided by by x minus one. 
So let's try plugging in number three 
into this function. 
So I plug in number three into this 
function and I have to just compute, 
right? 
Three squared minus one over three minus 
one well, that's three squared is nine 
minus one is eight three minus one is two 
and nine divided by two is four And sure 
enough, out of this function comes the 
number four Let's look at that example 
again but with a little bit more detail. 
this is actually a pretty complicated 
function. 
Alright? 
But I can open up the function. 
Alright. 
And take a look at how the functions 
actually doing its calculations. 
You can think of this function as having 
three different steps. 
Alright. 
One of the steps squares its input and 
subtracts one, and so I calculate the 
numerator. 
Another step just subtracts one from its 
input. 
The outputs of those two steps then get 
plugged into the division. 
And that's how I get the output of this 
big complicated function. 
Now, something like x^two - one, 
you could also think of that as having 
some, you know separate steps as well. 
But this is good for right now. 
Okay. 
Now let's see what happens. 
I take the number three and I plug it 
into the function. 
Alright? 
Now I'm going to be calculating the 
numerator and the denominator separately, 
so I'll take those 3s, and up here, I'll 
look at three^two - one and I'll get out 
eight. 
And down here, three - one became two Now 
the eight and the two get plugged into 
the division, and eight divided by two is 
four and that becomes the output of the 
function, 
right? 
Input's three, output is four but when I 
look at it this way, I can see how all 
the steps are, are playing out. 
Okay. 
I evaluate the function at three, but who 
cares? 
Well, let's try to evaluate the function 
at one instead of at three. 
So what happens when we plug in the 
number one into this function? 
I got the number one here. 
I'm going to look inside. 
I'm going to open up this function. 
Now imagine I've got this number one. 
I'm going to plug it into the function. 
All right. 
Now I'm going to be evaluating the 
numerator and denominator separately, so 
I'm going to take this one and split it 
up, and plug it into the numerator and 
the denominator. 
The numerator sends its input to its 
input squared minus one. 
So one^two minus one is zero and the same 
thing down here, one - one is zero Now 
I've got 0 and 0 which I'm going to be 
plugging in to the. 
Okay, very bad. 
Right? 
I'm dividing by zero and I can not 
proceed, so this function is not defined 
at one. 
So I can't plug one into the function. 
But if I wanted to figure out what the 
function's value was that inputs near 
one, I could do that. 
So let's try to plug in one point one 
instead so let's plug one point one into 
this function. 
I can't plug in one because I need to 
divide them by zero, but let's try 
plugging in one point one I'm going to 
open up the function again and take one 
point one plug it into the function. 
Now one point one is going to to be 
evaluated in the numerator and the 
denominator. 
one point one^second minus one is 
twentyone. 
And one point one minus one became one. 
Now twentyone and one are going into the 
division. 
And twentyone divided by one is two point 
one So when I evaluate the function at 
one pint one I get out two point one. 
Instead of just plugging in one value, 
let's plug in a whole bunch of values. 
We'll make a table. 
So use that same function again. 
F of x is x^second minus one divided by x 
minus one Now, I can't plug 1 into the 
function, 'because if I plug in one, I'd 
be dividing by zero, and I can't divide 
by zero. 
One isn't in the domain of this function. 
But I can plug in numbers near one, 
right? 
And we saw that one point one if I plug 
in that, I get two point one. 
Right? 
And if I plug in one point zero one I get 
two point zero one If I plug in 1 point 
zero zero one I get two point zero zero 
one. 
Right? 
And so on. 
If I plug in 1.000001 I get 2.000001. 
Right. 
Well, what's going on here? 
I could summarize this situation by 
saying the following. 
The limit of x squared minus one over x 
minus one as x approaches one is equal to 
two. 
Why is that? 
Well, this is because. 
I can make x^2-1 over x minus one. 
As close to two as I want. 
If. 
I make x. 
Close enough. 
To one. 
Lets see. 
Here's my table alright if you want the 
output of this function to be within a 
billionth of two all you need to do is to 
make sure that your input is within a 
trillionth of one alright. 
As long as your input is close enough to 
one you can guarantee that your output is 
as close to two as you like. 
This is just looking at a table of 
values. 
You know, maybe a dozen values and seeing 
what they're getting close to. 
It would be a lot better if there were a 
more convincing argument. 
So let's go back to our definition of 
limit. 
To say the limit of f of x equals l means 
that f of x can be made as close to l as 
you desire by making x close enough to a. 
And let me emphasize something. 
Close enough. 
But not equal. 
To a. 
Why does something like this matter? 
Well, let's go back to our example. 
In our example the function wasn't 
defined at one. 
But the limit doesn't depend upon the 
function's value at one. 
It only depends on the function's value 
near one. 
So x squared minus one over x minus one 
is equal to x plus one as long as x isn't 
equal to one right. 
As long as x isn't one this is a true 
statement. 
So now what's the limit as x goes to one 
of x squared minus one over x - one. 
Well, this is the limit as x approaches 
one of x. 
one = one. 
because the limit doesn't depend upon the 
value of the function at one. 
It only depends upon the values of the 
function near one. 
And as a result, these two things have 
the same limit. 
Even better the limit of x plus one, as x 
approaches one, well that's the limit of 
a sum. 
And the limit of a sum is the sum of the 
limits. 
So I can rewrite this limit as the limit 
as x goes to one of x plus the limit as x 
goes to one of one. 
And what the limit of x as x goes to one? 
Well that's asking what can I make x 
close to if I make x close enough to one? 
Well that's one. 
And the limit of one as x goes to one is 
asking me what's one close to when x is 
close to well there's not even an x in 
this right wiggling x doesn't affect this 
at all so that limits also one. 
And one plus one is two so indeed the 
limit. 
x^twenty two minus one divided by x - one 
as x approaches one is two. 
Limits provide information about what a 
functions values are approaching, 
alright. 
It's a way of accessing otherwise 
forbidden information. 
I might not be able to plug in the value 
one, because that would have entailed 
dividing by zero. 
And yet I know, that the functions output 
is as close to two as I like. 
As long as the input is close to but not 
equal to one.