Delta-Epsilon Proof: Determine Limit l for f(x) = x4

[LaMantequilla]

Homework Statement

Determine the limit l for the given a, and prove that it is the limit by showing how to find a ∂ such that |f(x)-l| < ε for all x satisfying o < |x-a| < ∂.

f(x) = x⁴ , arbitrary a

(Spivak's Calculus 5-3iv)

Homework Equations

The Attempt at a Solution

l = a⁴

The best I can do is show that |x² - a²| < ε for |x-a|<∂₁ = min(ε/(2|a|+1),1). After that, I get lost.

Since I found that |x² - a²| < ε for |x-a|<∂₁ = min(ε/(2|a|+1),1), can I rearrange |x⁴ - a⁴| to |(x²)² - (a²)²| to find that ∂₂=min(1,ε/(2|a|²+1)? This seems to be what Spivak's answer book is suggesting, but I'm not sure.

Can someone walk me through this? I'm pretty new to delta-epsilon proofs.

 

[Bacle2]

Have you considered the factoring:

a²-b²=(a+b)(a-b)?

It may help.

And ,

I heard it goes great with butter...

 

[Bacle2]

Sorry if the last comment above seemed weird: I was just making reference to

your name 'Mantequilla' .

 

[LaMantequilla]

Ahh, yes. It made me laugh when I read it. I would have responded earlier, but I've been terribly busy these past few days.

I'm still having trouble with this proof. Even though I have Spivak's answer, I just don't see how he arrives at it.

Using the (a²)²-(b²)² factoring, we end up with (a²-b²)(a²+b²). The (a²+b²) part is giving me the most trouble, since I can't do difference of squares.