Is this a suitable epsilon-delta proof of a limit?

Question

Fairly new to using $\epsilon$ and $\delta$ proofs so could someone tell me if what I've got in the attached image is a suitable way to prove $$\lim_{x \to 0} 3x^3 = 0$$ Cheers

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Thanks, I know using LaTeX is better.. bit pushed for time and only spent a couple of hours learning it! — Nate, May 15 '14 at 19:12

score 1 · Answer 1 · answered May 15 '14 at 19:19

It looks quite nice. Two comments:

It's good practice to clearly separate the actual proof (the part that starts with "Take $\delta < ( \frac{\varepsilon}{3} )^{\frac{1}{3}}$.") from the calculations that lead to your choice of $\delta$.
The last inequality in your scratch work is backwards. Specifically, $$ |x| < \left( \frac{\varepsilon}{3} \right)^{\frac{1}{3}} < \delta $$ should read $$ |x| < \left( \frac{\varepsilon}{3} \right)^{\frac{1}{3}} = \delta. $$ The whole point (as you seem to understand) is that when $|x| < \delta$, then $x$ is close enough to $0$. You can always make $\delta$ smaller and the proof will still be valid. But, you cannot make $\delta$ bigger without potentially compromising the proof.

Is this a suitable epsilon-delta proof of a limit?

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