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[MUSIC] So, what does it mean to say that 
the limit of x^2, as x approaches 2, is 

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4? Strictly speaking, it means that you 
can make x^2 as close as you want to 4 by 

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making x sufficiently close to 2, 
but talking in that way can be a little 

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bit confusing. 
So I'm going to sort of format it here 

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as, as if it were a dialogue. Alright? So 
you're going to make some sort of demand. 

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You're going to demand that x^2 be close 
to 4. 

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Maybe you're going to demand that x^2 be 
within 1/10 of 4. 

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Alright? So that means you're, you're 
asking that x^2 be between 3.9, 

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3.9 is a 1/10 less than 4, 
and 4.1, which is a 1/10 more than 4. 

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So you're going to make some demand that 
the output be close to 4 and I have to 

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satisfy your demand by making x 
sufficiently close to 2. 

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Now, what does that mean? I'm going to 
satisfy your demand by stipulating that x 

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be within some small distance of 2. 
So, how close is sufficiently close? 

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Well, in this case, let's make let's make 
x be within a 1/100 of 2 and see what 

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happens. 
So x is within a 1/100 of 2 and that 

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means that x [SOUND] is bigger than 1.99 
[SOUND] and smaller [SOUND] than 2.01. 

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And if x is bigger than 1.99, then x^2 is 
bigger than 3.9601, and if x is smaller 

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than 2.01, then x squared is smaller than 
[SOUND] 4.0401. 

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And, and look at these numbers, 
3.9601, that is bigger [SOUND] than 3.9, 

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and over here, 4.0401, that's smaller 
than 4.1. 

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So notice what happened here. 
If x is within a 1/100 of 2, then x is 

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between 1.99 and 2.01. 
But if x is between 1.99 and 2.01, then 

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x^2 is between 3.9601 and 4.0401, 
and if x^2 is bigger than 3.9601, it's 

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bigger than 3.9, 
and if x^2 is smaller than 4.0401, it's 

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smaller than 4.1. 
So that means demanding that x be within 

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a 1/100 of 2, in fact, forces x^2 to be 
between 3.9 and 4.1. 

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In other words, it forces x^2 to be 
within a 1/10 of 4. 

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So if your demand is that x squared be 
within a 1/10 of 4, I can satisfy that 

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demand by simply requiring x to be within 
a 1/100 of 2. 

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Now, I can do the same thing for other 
demands that you might make. 

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You might have demanded that x^2 be 
within a 1/100 of 4 and I can do that as 

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well. 
So that would mean that x^2 is between 

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3.99 and 4.01. 
Well, if x^2, if you want x^2 to be 

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between 3.99 and 4.01, I'm going to have 
to make some condition on how close x 

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have to be to 2. 
So let's try a 1/1000. 

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So if x is within a 1/1000 of 2, that 
means that X, it'll be between 1.999 and 

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2.001. 
And if x is bigger than 1.999, then x^2 

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is bigger [SOUND] than 3.996001. 
And if x < 2.001, then x^2 < 4.004001. 

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Now, look at these numbers, 3.996001, 
that is bigger 3.99, 

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and 4.004001, that is smaller than 4.01. 
So if x is within a 1/1000 of 2, that 

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means x is between 1.999 and 2.001. 
That means that x^2 is between 3.996 and 

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4.004, a little bit more. 
That makes x^2 be, between 3.99 and 4.01, 

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which is exactly what you demanded. 
So this is the structure of, of what it 

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means to say that the limit of x^2 is 4 
as x approaches 2. It means no matter 

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close you demand x ^ 2 to be to 4, even 
if this epsilon were replaced by a very 

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tiny number. 
I'd be able to find some other distance, 

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some number to put here, so that as long 
as x is that close to 2, then x^2 is as 

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close as you wanted to the number 4. 
So the exercise, below asks you to take a 

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look at another situation like this to 
see if you can figure out number to put 

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here to satisfy some condition that 
someone might make on you. 

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[MUSIC]