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We had to prove that

$\lim_{x \to 1} x^3 = 1$

Starting with

$|x^3 - 1| < \epsilon$

By breaking the inequality I got

$\sqrt[3]{1 - \epsilon} < x < \sqrt[3]{1 + \epsilon}$

I then put my delta as

$\delta = \min({\sqrt[3]{1 - \epsilon} - 1, \sqrt[3]{1 + \epsilon} - 1})$

This is the mistake on my part that cost me my mark, since the left expression should actually be

$1 -\sqrt[3]{1 - \epsilon}$

Because of this mistake, the left side is always negative, so the minimum will be always negative, so the proof wouldn't make sense, but if you use the right side you can derive the correct epsilon equation.

My question is, since $|x - 1| < \delta$, delta can't be negative because it's greater than an absolute value, so wouldn't the left expression never be true? Is this still a correct, if messy, epsilon delta proof?

CuriousMathPerson
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    Not following. $\delta $ should be a function of $\epsilon$, not a function of $x$. And where did the $1$ go? – lulu Feb 09 '24 at 19:02
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    Carefully write down the definition of a limit. That definition should clear up your question. – Xander Henderson Feb 09 '24 at 19:02
  • "Starting with..." - This has nothing to do with the definition of the limit. – Filippo Feb 09 '24 at 19:07
  • The x should have been a 1, I fixed it – CuriousMathPerson Feb 09 '24 at 19:08
  • Filippo that's the scratch work to find a suitable delta – CuriousMathPerson Feb 09 '24 at 19:09
  • Ted I think since delta is known to be greater than an absolute value, the negative value of the minimum could never be true, it would contradict the delta statement – CuriousMathPerson Feb 09 '24 at 19:28
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    No, you really need the minmum of two positive quantities to get a valid $\delta$. And your negative $\delta$ certainly disqualifies your argument. I would have given a bit of partial credit, but less than half. And you’ve only shown the scratchwork and not the official proof, so that also loses credit. … You can give me a formula for a negative $\delta$ and say “well, you know $\delta$ has to be positive” and expect to get credit. – Ted Shifrin Feb 09 '24 at 19:28

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