eplison delta proof problem

• October 23rd 2012, 07:45 AM
kingsolomonsgrave
eplison delta proof problem
Attachment 25363

Somehow the methodology never makes sense to me.

I understand the idea behind it, but I never understand how one chooses the appropriate number for epsilon. Can someone break down how one would do this problem?

thanks!
• October 23rd 2012, 08:17 AM
Plato
Re: eplison delta proof problem
Quote:

Originally Posted by kingsolomonsgrave
Attachment 25363
Somehow the methodology never makes sense to me.
I understand the idea behind it, but I never understand how one chooses the appropriate number for epsilon.

The fact is you do not choose the epsilon. That is given at the start of the process.
Your task is to a delta that makes it all work out.
So you control delta, in this case the size of $|x-3|.$
Look at $\left| {\frac{{2x + 4}}{5} - 2} \right| = \left| {\frac{{2x - 6}}{5}} \right| = \frac{2}{5}\left| {x - 3} \right|$.
Ah, there is that factor you control. To get $\frac{2}{5}\left| {x - 3} \right|<\varepsilon$ we pick $\delta=\frac{5\varepsilon}{2}$
• October 23rd 2012, 11:20 AM
hollywood
Re: eplison delta proof problem
Right - you're given $\epsilon$ and you need to find a $\delta$ (which will be a function of $\epsilon$) that works. So you're doing the algebra backwards. Then when you write your proof, doing the algebra forward, it works. There's an excellent paper on how to do these proofs right here on the MHF Calculus forum - it's one of the "sticky" threads at the top.

- Hollywood