# Thread: Precise definition of a limit question

1. ## Precise definition of a limit question

For the limit
lim
x → 2
(x^3 − 4x + 7) = 7

illustrate the definition by finding the largest possible values of δ that correspond to ε = 0.2 and ε = 0.1. (Round your answers to four decimal places.)...note:let d=delta, E=epsilon

My work:
0</x-a/<d, then /f(x)-L/<E
0</x-2/<d, then /x^3-4x/<E...

so taking the right part of the statement further:

/x(x^2-4)/<E
/x(x-2)(x+2)/<E
/x-2/< E/(x^2+2x)

d<E/(x^2+2x)

I'm not sure what to do here. Even if I plug in E=0.2, I'm still left with the x's...please help me out, any advice/tips are appreciated.

2. ## Re: Precise definition of a limit question

What you have done is good, but $\epsilon$ or $\delta$ can't depending of the variable $x$.

If we define the limit with the $\epsilon-\delta$ definition:
$0<|x-2|<\delta \Rightarrow |f(x)-7|<\epsilon$

We calculate:
$|f(x)-7|=|x^3-4x+7-7|=|x^3-4x|$ and $|x^3-4x|<\epsilon \Leftrightarrow |x||x+2||x-2|<\epsilon$

Now define $\delta = 1$ therefore $|x-2|<1$ (we want $x$ close to 2).
Determine $|x|$ and $|x+2|$ with this condition.

But have you ever done examples like this before?
Also I recommand you to read the sticky thread of Krizalid in forum calculus about epsilon-delta proofs.

3. ## Re: Precise definition of a limit question

How do you define 'd' as =1? I think I missed something...would you mind explaining this?

so /x-2/<1
so 2<x<3
and /x/<1
-1<x<1

is this part right?? You caught me, this is my first time doing a problem like this...Strangely my book has no examples of this type of prob. Just" Prove" problems...If I found an example, I'd be all set. I'll go try to find that thread and read through it! But thanks for your help!

4. ## Re: Precise definition of a limit question

Like I said before, it can be useful to take a look at the sticky thread of Krizalid in forum calculus about epsilon-delta proofs. In this document there're a lot of examples:

http://www.mathhelpforum.com/math-he...ofs-47767.html