Epsilon-delta confusion

[#1procrastinator]

Prove that lim x->3 x^2 = 9

when

The book factors the expression



Then it says that if we can find a positive constant C, then we can write



It then goes on to say we can make by taking

^ What does it mean 'make ' by moving the constant over to the other side? If they wrote it like that, then doesn't it already mean that they've chosen some C such that the expression on the LHS is less than epsilon?

Also, I don't understand the motivation for finding the constant C and doing the above. Is it just to try and get the expression into the form ?

Then after that, it says we can find a number C if we restrict x to lie in some interval centered at 3. Since we are interested only in values that are close to 3, it is reasonable to assume that x is within a distance 1 from 3' - is that just a random small number they chose? is 1 the standard to be chosen in similar limit problems?

So that leads to

Then that leads to , thus . Does that expression mean that we only care that is less than , not what the valid values of x are?

So then they choose remembering that but there are two restrictions on

and

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So ….they show that the works and they choose and get





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so for the limit to work, you need to use 2 different deltas simulatenously or something? this is rather confusing as the steps seem a bit random to me

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also, when they prove , and they use , they did


^ is simply because and the inequality's just been multiplied by 4?

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[kamil9876]

It then goes on to say we can make by taking

^ What does it mean 'make ' by moving the constant over to the other side? If they wrote it like that, then doesn't it already mean that they've chosen some C such that the expression on the LHS is less than epsilon?

What it means is that every which satisfies also satisfies (They are working backwards, so they are assuming that such a  was already found)

Also, I don't understand the motivation for finding the constant C and doing the above. Is it just to try and get the expression into the form |?

Yeah pretty much, often in Analysis it's good to try to replace some garbage like |x+3| with a constant to makes things easier

Then after that, it says we can find a number C if we restrict x to lie in some interval centered at 3. Since we are interested only in values that are close to 3, it is reasonable to assume that x is within a distance 1 from 3' - is that just a random small number they chose? is 1 the standard to be chosen in similar limit problems?

Well sometimes you need to choose something even smaller, it depends on what sort of inequality you want: In general, the tighter the inquality you want the smaller the interval you want to choose.

Anyway if it helps here is a summary of the argument:

Fix an we want to find some small enough interval around so that for all in that interval. There is some constant so that for close enough to we have that (the details of how small this interval is and what we can figure out later). Let's call this interval , so we see that . Now we can find an even smaller interval so that if then (again details of what I' is is left to the reader).
So now we see that if then:

(since because )

(since )



The rest is just finding such and so we know it does indeed exists.

[kamil9876]

I've spent my time looking for this classic, it pretty much explains all you need to know:

[JinXi]

I've spent my time looking for this classic, it pretty much explains all you need to know:

(Image removed from quote.)

ahaha best math joke ever 

[#1procrastinator]

Thanks a lot kamil