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[MUSIC] Limits are probably the most 
important concept in this course. 

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So we should really have a definition of 
what we mean by limit. 

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Now here is what we mean by limits. 
To say that the limit of f of x as x 

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approaches a is equal to L means that f 
of x can be as close to L as desired by 

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making x close enough to a. 
There is a tons of subtlety to this 

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definition so it's worth to look at an 
example. 

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So let's take a look at this function. 
This is the function that takes an input 

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x and spits out x^second minus one 
divided by by x minus one. 

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So let's try plugging in number three 
into this function. 

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So I plug in number three into this 
function and I have to just compute, 

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right? 
Three squared minus one over three minus 

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one well, that's three squared is nine 
minus one is eight three minus one is two 

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and nine divided by two is four And sure 
enough, out of this function comes the 

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number four Let's look at that example 
again but with a little bit more detail. 

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this is actually a pretty complicated 
function. 

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Alright? 
But I can open up the function. 

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Alright. 
And take a look at how the functions 

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actually doing its calculations. 
You can think of this function as having 

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three different steps. 
Alright. 

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One of the steps squares its input and 
subtracts one, and so I calculate the 

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numerator. 
Another step just subtracts one from its 

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input. 
The outputs of those two steps then get 

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plugged into the division. 
And that's how I get the output of this 

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big complicated function. 
Now, something like x^two - one, 

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you could also think of that as having 
some, you know separate steps as well. 

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But this is good for right now. 
Okay. 

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Now let's see what happens. 
I take the number three and I plug it 

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into the function. 
Alright? 

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Now I'm going to be calculating the 
numerator and the denominator separately, 

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so I'll take those 3s, and up here, I'll 
look at three^two - one and I'll get out 

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eight. 
And down here, three - one became two Now 

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the eight and the two get plugged into 
the division, and eight divided by two is 

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four and that becomes the output of the 
function, 

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right? 
Input's three, output is four but when I 

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look at it this way, I can see how all 
the steps are, are playing out. 

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Okay. 
I evaluate the function at three, but who 

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cares? 
Well, let's try to evaluate the function 

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at one instead of at three. 
So what happens when we plug in the 

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number one into this function? 
I got the number one here. 

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I'm going to look inside. 
I'm going to open up this function. 

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Now imagine I've got this number one. 
I'm going to plug it into the function. 

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All right. 
Now I'm going to be evaluating the 

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numerator and denominator separately, so 
I'm going to take this one and split it 

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up, and plug it into the numerator and 
the denominator. 

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The numerator sends its input to its 
input squared minus one. 

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So one^two minus one is zero and the same 
thing down here, one - one is zero Now 

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I've got 0 and 0 which I'm going to be 
plugging in to the. 

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Okay, very bad. 
Right? 

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I'm dividing by zero and I can not 
proceed, so this function is not defined 

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at one. 
So I can't plug one into the function. 

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But if I wanted to figure out what the 
function's value was that inputs near 

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one, I could do that. 
So let's try to plug in one point one 

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instead so let's plug one point one into 
this function. 

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I can't plug in one because I need to 
divide them by zero, but let's try 

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plugging in one point one I'm going to 
open up the function again and take one 

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point one plug it into the function. 
Now one point one is going to to be 

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evaluated in the numerator and the 
denominator. 

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one point one^second minus one is 
twentyone. 

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And one point one minus one became one. 
Now twentyone and one are going into the 

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division. 
And twentyone divided by one is two point 

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one So when I evaluate the function at 
one pint one I get out two point one. 

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Instead of just plugging in one value, 
let's plug in a whole bunch of values. 

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We'll make a table. 
So use that same function again. 

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F of x is x^second minus one divided by x 
minus one Now, I can't plug 1 into the 

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function, 'because if I plug in one, I'd 
be dividing by zero, and I can't divide 

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by zero. 
One isn't in the domain of this function. 

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But I can plug in numbers near one, 
right? 

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And we saw that one point one if I plug 
in that, I get two point one. 

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Right? 
And if I plug in one point zero one I get 

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two point zero one If I plug in 1 point 
zero zero one I get two point zero zero 

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one. 
Right? 

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And so on. 
If I plug in 1.000001 I get 2.000001. 

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Right. 
Well, what's going on here? 

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I could summarize this situation by 
saying the following. 

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The limit of x squared minus one over x 
minus one as x approaches one is equal to 

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two. 
Why is that? 

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Well, this is because. 
I can make x^2-1 over x minus one. 

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As close to two as I want. 
If. 

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I make x. 
Close enough. 

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To one. 
Lets see. 

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Here's my table alright if you want the 
output of this function to be within a 

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billionth of two all you need to do is to 
make sure that your input is within a 

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trillionth of one alright. 
As long as your input is close enough to 

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one you can guarantee that your output is 
as close to two as you like. 

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This is just looking at a table of 
values. 

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You know, maybe a dozen values and seeing 
what they're getting close to. 

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It would be a lot better if there were a 
more convincing argument. 

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So let's go back to our definition of 
limit. 

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To say the limit of f of x equals l means 
that f of x can be made as close to l as 

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you desire by making x close enough to a. 
And let me emphasize something. 

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Close enough. 
But not equal. 

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To a. 
Why does something like this matter? 

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Well, let's go back to our example. 
In our example the function wasn't 

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defined at one. 
But the limit doesn't depend upon the 

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function's value at one. 
It only depends on the function's value 

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near one. 
So x squared minus one over x minus one 

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is equal to x plus one as long as x isn't 
equal to one right. 

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As long as x isn't one this is a true 
statement. 

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So now what's the limit as x goes to one 
of x squared minus one over x - one. 

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Well, this is the limit as x approaches 
one of x. 

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one = one. 
because the limit doesn't depend upon the 

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value of the function at one. 
It only depends upon the values of the 

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function near one. 
And as a result, these two things have 

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the same limit. 
Even better the limit of x plus one, as x 

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approaches one, well that's the limit of 
a sum. 

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And the limit of a sum is the sum of the 
limits. 

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So I can rewrite this limit as the limit 
as x goes to one of x plus the limit as x 

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goes to one of one. 
And what the limit of x as x goes to one? 

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Well that's asking what can I make x 
close to if I make x close enough to one? 

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Well that's one. 
And the limit of one as x goes to one is 

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asking me what's one close to when x is 
close to well there's not even an x in 

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this right wiggling x doesn't affect this 
at all so that limits also one. 

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And one plus one is two so indeed the 
limit. 

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x^twenty two minus one divided by x - one 
as x approaches one is two. 

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Limits provide information about what a 
functions values are approaching, 

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alright. 
It's a way of accessing otherwise 

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forbidden information. 
I might not be able to plug in the value 

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one, because that would have entailed 
dividing by zero. 

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And yet I know, that the functions output 
is as close to two as I like. 

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As long as the input is close to but not 
equal to one. 

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