(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that : (x^4+y^4)/(x^2+y^2) < ε if 0 < x^2 + y^2 < δ^2 for a suitably chosen δ depending on ε.

2. Relevant equations

[itex]\forall[/itex]ε>0, [itex]\exists[/itex]δ>0 | 0 < (x^2 + y^2)^(1/2) < δ [itex]\Rightarrow[/itex] |f(x,y) - L| < ε

Obviously here were dealing with lim (x,y)→(0,0) f(x,y) = 0 so the following statement is equivalent and more convenient to use in my opinion:

[itex]\forall[/itex]ε>0, [itex]\exists[/itex]δ>0 | 0 < |x|,|y| < δ [itex]\Rightarrow[/itex] |f(x,y) - L| < ε

3. The attempt at a solution

So we know : |x| < δ [itex]\Rightarrow[/itex] x^2<δ^2 and also |y|<δ [itex]\Rightarrow[/itex] y^2<δ^2

And using the triangle inequality we also consider : |x^4 + y^4| ≤ |x|^4 + |y|^4

So putting those together we observe :

|f(x,y) - L| = |(x^4+y^4)/(x^2+y^2)| ≤ (|x|^4 + |y|^4)/(|x|^2 + |y|^2) < 2δ^4/2δ^2 = δ^2 ≤ ε

[itex]\Rightarrow[/itex] δ = [itex]\sqrt{ε}[/itex]

Now that I have my δ, I could go through and prove that it was the right δ, but I have one problem. The book says that δ = [itex]\sqrt{ε/2}[/itex] so I'm wondering where I went wrong or is this a typo in the book? If it helps I also tried using the other statement 0 < (x^2+y^2)^(1/2) < δ and got the right answer, but I'm not sure why I'm wrong about this other method?

Thanks.

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# Homework Help: Multivariate ε-δ proof

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