Precise Definition of a Limit

[JasonRox]

How do I find g?

It's so confusing.

I'm trying to learn this on my own, so bare with me.

I'm going with an example I know the answer to, and maybe someone can work with me here. I'll ask questions through the solution.

We will do x^3 since that is complicated enough, but I understand the steps, just not the logic to moving on to the next step.

|x^3-a^3|<e
0<|x-a|<g

Find a value for g that satisfies the above.

|x^3-a^3|=|x-a||x^2+ax+a^2|
|x^2+ax+a^2|<=|x|^2+|a||x|+|a|^2

Note: |x|-|a|<=|x-a|<1
If you don't understand why one is chosen maybe this isn't for you. In case you forgot, we take 1 because that guarantees that the difference won't be too big.

|x|<1+|a|

Take the above and you get...

[tex](1+|a|)^2+|a|(1+|a|)+|a|^2

WARNING: This is in the works. I will be back to complete it.

[matt grime]

You'd be better using that |a|< |x| +g

or better yet, assuming that g is chosen such that |a|<2|x|, since if some g works, a smaller g has to work too, so there's no harm in placing a maximal size on g that helps eliminate a (assuming x is not zero. if x is zero it's quite easy)