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

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Homework Help Overview

The discussion revolves around determining the limit \( l \) for the function \( f(x) = x^4 \) as \( x \) approaches an arbitrary value \( a \). The original poster is tasked with proving this limit using a delta-epsilon argument, specifically showing how to find a \( \delta \) such that \( |f(x)-l| < \epsilon \) for all \( x \) satisfying \( 0 < |x-a| < \delta \).

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to show that \( |x^2 - a^2| < \epsilon \) for a specific \( \delta \) but expresses uncertainty about the next steps. They consider rearranging \( |x^4 - a^4| \) using the difference of squares but are unsure about the implications of this approach.

Discussion Status

Participants are exploring different factoring techniques to assist in the proof. One participant suggests using the factoring \( a^2 - b^2 = (a+b)(a-b) \), which may provide a pathway to simplify the expression. The original poster acknowledges difficulty in understanding how to apply this factoring effectively, particularly with the term \( a^2 + b^2 \).

Contextual Notes

The original poster indicates they are new to delta-epsilon proofs and are seeking guidance through the process. There is a reference to Spivak's answer book, suggesting that the original poster is trying to reconcile their understanding with provided solutions.

LaMantequilla
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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.
 
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Have you considered the factoring:

a2-b2=(a+b)(a-b)?

It may help.

And ,

I heard it goes great with butter...
 
Sorry if the last comment above seemed weird: I was just making reference to

your name 'Mantequilla' .
 
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.
 

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