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

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

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

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

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

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

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

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

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

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

satisfy your demand by making x sufficiently close to 2.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

[MUSIC]