1. ## Strange Limit Proof

I thought I understood how to show that a function has a limit using the limit definition, but my book is doing a problem in a way I can't follow. The question is "prove that the limit of x^2 as x approaches 2 is 4"
The book starts with the epsilon equation
$\displaystyle \left| {x^2 - 4} \right| < \varepsilon$

and they show that
$\displaystyle \sqrt {4 - \varepsilon } < x < \sqrt {4 + \varepsilon }$
which I can understand (though somehow they assume epsilon is less than 4)

Then they say that they want to find delta. They show that delta is
$\displaystyle Min\{ 2 - \sqrt {4 - \varepsilon } ,\sqrt {4 + \varepsilon } - 2\}$
Which I am now able to follow although it took awhile to see why this is true. Up to this point I can understand what is going on. This is all the "scratch work" if I am correct. Now they have to actually prove the limit and they say the following...

"If delta has this or any smaller positive value, the inequality
0<abs(x-2)<delta
will place x between
$\displaystyle \sqrt {4 - \varepsilon } \_and\_\sqrt {4 + \varepsilon }$
This completes the proof."

I don't understand how this completes the proof. I can prove the limit a different way but it is disheartening that I can't follow this. Can anyone explain how their work proves the limit. Thanks.

2. Originally Posted by billa
I thought I understood how to show that a function has a limit using the limit definition, but my book is doing a problem in a way I can't follow. The question is "prove that the limit of x^2 as x approaches 2 is 4"
The book starts with the epsilon equation
$\displaystyle \left| {x^2 - 4} \right| < \varepsilon$

and they show that
$\displaystyle \sqrt {4 - \varepsilon } < x < \sqrt {4 + \varepsilon }$
which I can understand (though somehow they assume epsilon is less than 4)

Then they say that they want to find delta. They show that delta is
$\displaystyle Min\{ 2 - \sqrt {4 - \varepsilon } ,\sqrt {4 + \varepsilon } - 2\}$
Which I am now able to follow although it took awhile to see why this is true. Up to this point I can understand what is going on. This is all the "scratch work" if I am correct. Now they have to actually prove the limit and they say the following...

"If delta has this or any smaller positive value, the inequality
0<abs(x-2)<delta
will place x between
$\displaystyle \sqrt {4 - \varepsilon } \_and\_\sqrt {4 + \varepsilon }$
This completes the proof."

I don't understand how this completes the proof. I can prove the limit a different way but it is disheartening that I can't follow this. Can anyone explain how their work proves the limit. Thanks.
Assuming I understand you correctly proving that works because

$\displaystyle |x^2-4|<\varepsilon\implies -\varepsilon<x^2-4<\varepsilon$

Etc. Is that what you are asking?

3. umm no lol, but ty still. What i don't understand is given the books definition of delta as the minimum of two expressions, why is this statement true

0<abs(x-2)<delta
will place x between

and if that statement is true, why does it prove the limit?

4. A common definition for the limit is $\displaystyle \lim_{x\to a}f(x) = b \textrm{ iff } \forall \epsilon>0, \exists \delta>0 \textrm{ such that } |x-a|<\delta \implies |f(x)-b|<\epsilon$

They show that delta is
This is a definition of $\displaystyle \delta$ that exists and is >0 whenever $\displaystyle \epsilon>0$. So all we need to show now is that this definition of $\displaystyle \delta$ satisfies $\displaystyle |x-2|<\delta \implies |x^2-4|<\epsilon$ You can usually do this by reading how you got the definition for $\displaystyle \delta$ backwards, although this does not constitute a formal proof. To do it formally,

Assume x>2 so we can replace |x-2| with x-2.
$\displaystyle 0<x-2<min(2-\sqrt{4-\epsilon},\sqrt{4+\epsilon}-2)$
$\displaystyle 2<x<min(4-\sqrt{4-\epsilon},\sqrt{4+\epsilon})$
Find the minimum (Warning: the following is not standard notation)
$\displaystyle 4-\sqrt{4-\epsilon}?\sqrt{4+\epsilon}$
both of these are now >0 so we can safely square
$\displaystyle 16-8\sqrt{4-\epsilon}+4-\epsilon?4+\epsilon$
$\displaystyle 16-2\epsilon?8\sqrt{4-\epsilon}$
Still both >0
$\displaystyle 256-64\epsilon+4\epsilon^2?64(4-\epsilon)$
$\displaystyle 256-64\epsilon+4\epsilon^2?256-64\epsilon$
so ?=>
and so $\displaystyle \sqrt{4+\epsilon}$ is the minimum, so
$\displaystyle 2<x<\sqrt{4+\epsilon}$

Repeating this process assuming x<2 produces $\displaystyle \sqrt{4-\epsilon}<x<2$
so we can say that $\displaystyle \sqrt{4-\epsilon}<x<\sqrt{4+\epsilon}$

It is very easy to show that this is equivalent to $\displaystyle |x^2-4|<\epsilon$

though somehow they assume epsilon is less than 4
When dealing with limits you only need to worry about the small values of epsilon, as how the function behaves when x = 9 doesn't affect the limit as $\displaystyle x \to 1$