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[SOUND] Fundamentally, limits are 
promises. 

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When I tell you the limit of F of X 
equals L, as X approaches A; I'm 

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promising you something. 
I'm promising you that I can get F of X 

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as close to L as you like as long as X is 
close enough to A. 

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Thinking of limits as promises helps us 
to understand statements like these. 

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Let's suppose that you tell me that you 
know the limit of f of x as x approaches 

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a is equal to something, maybe this is 
equal to l. 

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And maybe the limit of g of x as x 
approaches a is equal to m. 

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What that really is, is a promise that 
you can make F of X as close to L as I 

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like. 
And it's a promise that you can make G of 

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X as close to M as I like, as long as I'm 
willing to make X close enough to A. 

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But if you can promise me that you can 
make F of X close to L, and G of X close 

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to M. 
Then I can turn back and promise you that 

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F of X plus G of X is as close to L plus 
M as you like. 

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Alright? 
If I want to make this close to 

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something, I just ask you to make F of X 
close enough to L, and G of X close 

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enough to M. 
So that F of X plus G of X, is as you 

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like to L plus M. 
In other words, the limit of a sum is the 

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

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difference of the limits. 
And the limit of a product is the product 

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of the limits. 
And what about quotient. 

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And something similar is true for 
division. 

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If the limit of F of X as X approaches A 
is L. 

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And the limit of G of X as X approaches A 
is M, which isn't zero, then the limit of 

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F of X over G of X, as X approaches A is 
L over M. 

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In other words, the limit of the quotient 
is the quotient of the limits, provided 

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those limits exist and, the limit of the 
denominator is non-zero. 

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Let us do something with our new found 
knowledge about limits of quotients. 

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Here's a limit problem: I'm going to 
limit x-squared over x plus one as x 

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approaches two, alright? 
I'll promise you that x-squared over x 

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plus one is close to something whenever x 
is close enough to two. 

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This is the limit of a quotient, and the 
limit of the quotient's the quotient of 

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the limits, provided the limit of the 
denominator is not zero, and in this 

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case, it's not. 
So the limit of the quotient is the 

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quotient of the limits. 
[SOUND] Here's the limit of the 

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numerator. 
The limit is X approaches two of X plus 

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one. 
This is the limit of the denominator. 

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Now this is the limit of x squared. 
X squared is X times X. 

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This is a limit of a product. 
And the limit of a products the product 

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to the limits so I can replace the limit 
of the numerator with a limit of X as X 

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approaches two times the limit of X as X 
approaches two. 

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because this is the limit of X times X 
and here's the product limits. 

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Limit of X times the limit of X as X 
approaches two. 

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The denominator here is the limit of X 
plus one as X approaches two but that's a 

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

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So the limit of X plus one [SOUND] is the 
limit of X plus the limit of one, as X 

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approaches two. 
Lets keep going. 

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So I've got the limit of X times the 
limit of X over the limit of X plus the 

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limit of one. 
And all of these limits are being taken, 

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as X approaches two. 
What's the limit of X? 

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That's asking what can you guarantee X is 
close to if you're willing to have X be 

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close enough to two? 
Two is the limit of X as X goes to two. 

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So limit of X as X goes to two is two. 
The limit of X as X goes to two is doubt 

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is for multiplying, divided by limit of X 
as X approaches two plus with the limit 

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of one as X approaches two. 
This is asking, what can I guarantee one 

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is close to two if I am willing to have X 
be close enough to two. 

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Well, one is already close to one, right. 
The limit of a constant is that constant. 

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So this is just one. 
Two times two is four. 

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Two plus one is three. 
And so the limit of this expression is 

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four-thirds. 
At this point you are asking yourself why 

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is the rule for limits of quotients 
different than limits of the products. 

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A limit of a product is a product of the 
limits. 

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Laws of limits exist. 
Why do I have to worry about the limit of 

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a nominator being non-zero, when I'm 
taking the limit of a quotient? 

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Most basically the problem is that you 
can't divide by zero. 

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You can't go around telling people that 
the limit of a quotient is a quotient of 

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the limits because the limit of the 
denominator might be zero and then you'd 

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be telling people to divide by zero, 
which they can't do. 

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You can't divide by zero. 
But you can think about it even a little 

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more subtlety. 
You know, let's kind of unpack this a 

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bit. 
Here's an example to think about: the 

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limit of x over x minus three as x 
approaches six. 

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This is no problem, alright? 
The numerator is a number close to six, 

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it's how we're thinking about it, and the 
denominator is a number close to six 

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minus three. 
A number close to six minus three, that 

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means the denominator is a number close 
to three. 

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Now, we've got a number close to six 
divided by a number close to three. 

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Well, that's a number close to two. 
And, indeed, I mean, this limit is equal 

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to two. 
I can make this quotient as close to two 

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as I like. 
Because I can make the numerator as close 

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to six as I need, the denominator is as 
close to three as I need, to guarantee 

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that this ratio is as close to two as you 
like. 

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So that limit is two. 
But what if instead of asking about the 

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limit as x approaches six, I'd ask about 
the limit as x approaches three. 

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Well, then what would I know? 
Then I'd know that the numerator was a 

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number close to three, and the 
denominator was a number close to three 

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minus three. 
The denominators aren't close to zero. 

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The limited denominator is zero. 
That's exactly the scenario that the rule 

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for taking limits of quotients is 
forbidding us from considering. 

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We're not allowed to use the rule for 
limits of quotients here because the 

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limited denominator is zero. 
But what really goes wrong? 

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I mean, yeah, I can't divide by zero. 
Fine, I'm not going to divide by zero, 

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I'm just dividing by numbers close to 
zero. 

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But what happens when I divide by numbers 
close to zero? 

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A number close to three divided by a 
number close to zero. 

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Is that close to anything? 
If a number's close to zero it might be 

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positive, and very small. 
Three divided by a small positive number 

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is a huge positive number. 
What if the denominator were a number 

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close to zero but negative? 
Very small, negative number close to 

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zero. 
Three divided by a small but negative 

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number? 
That would be a hugely negative number. 

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Every negative. 
Well, this means this thing should be 

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getting close to both a gigantic positive 
number and a very, very negative number. 

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This thing isn't getting close to 
anything. 

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Fundamentally that's the problem but not 
all is lost. 

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Think about this slightly harder example: 
the limit of x-squared minus one over x 

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minus one as x approaches one. 
It's a limit of a quotient. 

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So your first temptation is to replace 
the limit of a quotient, by the quotient 

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of the limits. 
But you can't replace it without quotient 

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of limits, because the limit of the 
denominator is zero. 

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The limit of X-1 as X approaches one is 
equal to zero. 

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So it seems like our limit laws have 
failed us. 

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And then we have got one more trick up 
our sleeve. 

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Look at the numerator. 
X squared - one that factors as X+1 

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times, X-1. 
First record that fact. 

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And I am going to record the fact that 
the numerator factor is X+1 times X-1. 

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Now I've still got a limit of a quotient 
and the limit of the denominator is still 

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zero. 
So it seems like we're stuck. 

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But now I've got a factor of X minus one 
in the numerator and a factor of X minus 

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one in the denominator, and I can use 
that fact. 

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What I want to do is imagine canceling 
these, right. 

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I'd like to write that this is equal to 
the limit of just X plus one, as X goes 

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to one. 
But note, these are not actually the same 

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function. 
Alright? 

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This thing up here is not defined at one, 
this thing is defined at one. 

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And yet the limit doesn't care. 
The limit only depends upon values of the 

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function near one. 
And near one, this and this are exactly 

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the same. 
I'm not allowed to plug one into this, I 

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am allowed to plug one into this. 
They're different functions. 

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But those functions are equal if you're 
only considering values that are near and 

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not equal to one. 
As a result, these limits are the same. 

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This is another limit of a sum, and the 
limit of a sum is the sum of the limits, 

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so this is the limit of X plus the limit 
of one as X approaches one. 

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The limit of X as X approaches one is 
one. 

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The limit of one, which is a constant, as 
X approaches one is one. 

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This is one plus one. 
This is two. 

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And so that limit is equal to two and our 
limit laws have again saved the day. 

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[SOUND]