Understanding Epsilon-Delta Limits for Solving Proofs

[Pete.Co.Lust]

Homework Statement

I want to ask a question about Epsilon-Delta definition...
I have already read a tons of definitions about Epsilon-Delta limit proof. But i am still stucking in some places...
E.g.) Prove it by using Espilon-Delta method, lim (x->4) x^3=64

Homework Equations

|x-4|<delta delta>0
|x^3|<espilon espilon>0

The Attempt at a Solution

I am kinda know what I should do, I changed |x^3-64| into |(x-4)|(x^2+4x+16)| and my objective afterward is to ensure| (x^2+4x+16)| is "small".
But i just don't know what I should do afterwards... Should I just Sub x=4 and then find out |(x^2+4x+16)|<48?

Plx help me :( ThX! :)

 

[arkajad]

The point is not in showing that | (x^2+4x+16)| is "small, but that it is "bounded" when x is close to 4. So, restrict yourself to the region, for instance, 3<x<5. What can be said about | (x^2+4x+16)| in this region? Is it always smaller than some fixed number?

 

[madah12]

This is very similar to the problem I just asked maybe the replies will help you ,here is the link https://www.physicsforums.com/showthread.php?t=430952

 

[Pete.Co.Lust]

arkajad,
Alright I got it. So u mean is to find out the max. number when x is "close" to 4? Afterwards we can state that |x-4|<1 (Because x is "close" to 4) and we can find the max. number of x is 5. we subsitute 5 into the equation and we have (5)^2+(4)(5)+16=61. And it would be espilon/61 afterwards?

If its espilon/61, we write this "delta=min{1,e/61}"? What does "delta=min{1,e/61}" actually means? ThankS

madah12,
The link is extremely useful for me to solve this problem. I found out the "solution"
(Which i don't know its right or not...) right there. Many thanks.

 

[arkajad]

min{1,e/61} means eiher 1 or e/61, whichever happens to be smaller.

 

[Pete.Co.Lust]

min{1,e/61} means eiher 1 or e/61, whichever happens to be smaller.

So what do u mean is that if i pick e=62, the output will be 1 and if i pick e=60, the output will be 60/61? min{1,e/61}

 

[arkajad]

Yes, that's it. But are interested in "however small is epsilon" part, and for epsilon small enough (e<61) you will never use 1 as an output.

 

[Pete.Co.Lust]

Ohhhh you I got it! Thx for the explanation! It helped me muCH!