How do I express delta as a function of epsilon in epsilon-delta proofs?

[Laura1321412]

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Homework Statement

Im having trouble with epsilon -delta proofs in general... I keep looking at lots of examples I am just having trouble figuring what steps to take first. My question is the lim (x,y) -- > (0,0) of x^4-y^4/ x^2+ y^2

I understand the definition of the epsilon delta proofs, i just can't wrap my brain around what to do!

Homework Equations

The Attempt at a Solution

Not much. I know i can figure out |x^4|/ x^2 +0 = |x| <= e , and then the same for the y component ... But I am not really sure why you can just let y or x =0 ? Super confused...

 

[gerben]

You need to show that for any value $$\varepsilon>0$$ there is a $$\delta$$ such that when the distance between (x,y) and (0,0) is smaller than $$\delta$$ the value of
$$|x^4-y^4/ x^2+ y^2|$$ is smaller than $$\varepsilon$$.

 

[Laura1321412]

Any suggestion on how i should start?

 

[losiu99]

How about factorising the numerator?

 

[Laura1321412]

okay,,,

so i did

(x^2 +y ^2 ) (x^2 - y ^2) and cancelled,

so |x+y||x-y| < e

am i able to put delta, as e/|x-y| , or can i only bring real numbers back and forth between the inequality signs?

 

[gerben]

Your question is now:
"My question is the lim (x,y) -- > (0,0) of x^2-y^2 "

You cannot use x and y to express delta.
Whatever epsilon I give you, you need to be able to give me some delta so that whenever the distance between (x,y) and (0,0) is smaller than delta the distance between x^2-y^2 and the limit L, is smaller than epsilon. So, you better express delta as a function of epsilon, that way you will have an answer for any epsilon I might give you.