Epsilon-Delta Proof Homework: Understanding the Thought Process

[dainty77]

Homework Statement

For every ε> 0, there is a δ> 0 such that 1- δ< x <1 + implies 2- ε <7-5x <2 + ε

Homework Equations

The Attempt at a Solution

My understanding of epsilon-delta proofs is very minimal at this point. Was hoping someone would be able to explain your thought process when attempting these kinds of proofs. Thank you!

 

[tiny-tim]

hi dainty77!

Was hoping someone would be able to explain your thought process when attempting these kinds of proofs.

my thought process would be to attempt the simplest possible solution first …

ie try δ = kε first (for a constant k),

then δ = kε², then δ = k√ε and so on

 

[HallsofIvy]

I would start with $2- \epsilon< 7- 5x$ and see that $-5- \epsilon< -5x$ so that $x< 1+ \epsilon/5$ and $x- 1< \epsilon/5$. Then turn to $7- 5x< 2+ \epsilon$, so that $-5x< -5+ \epsilon$ and $x> 1- \epsilon/5$, $x- 1> -\epsilon/5$. Now do you see what $\delta$ must be?

 

[dainty77]

hi dainty77!


my thought process would be to attempt the simplest possible solution first …

ie try δ = kε first (for a constant k),

then δ = kε², then δ = k√ε and so on

I see where you are gettiing at! Let me work on it some more. Thank you for your help!