- #1
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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.
[tex]|x^3-a^3|<e[/tex]
[tex]0<|x-a|<g[/tex]
Find a value for g that satisfies the above.
[tex]|x^3-a^3|=|x-a||x^2+ax+a^2|[/tex]
[tex]|x^2+ax+a^2|<=|x|^2+|a||x|+|a|^2[/tex]
Note: [tex]|x|-|a|<=|x-a|<1[/tex]
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.
[tex]|x|<1+|a|[/tex]
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.
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.
[tex]|x^3-a^3|<e[/tex]
[tex]0<|x-a|<g[/tex]
Find a value for g that satisfies the above.
[tex]|x^3-a^3|=|x-a||x^2+ax+a^2|[/tex]
[tex]|x^2+ax+a^2|<=|x|^2+|a||x|+|a|^2[/tex]
Note: [tex]|x|-|a|<=|x-a|<1[/tex]
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.
[tex]|x|<1+|a|[/tex]
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.