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

Prove that as x --> c, lim (x^2) = (c^2) using only the definition. What does this tell you about x --> c, lim (x^3) = (c^3)? x --> c, lim (x^4) = (c^4) ? Prove it.

2. Relevant equations

The definition of a limit.

3. The attempt at a solution

Let $\epsilon > 0$ be given and let $\delta=\sqrt{c^2 + \epsilon} - c$. Then, if $0 < |x - c| < \delta$, then

$|x^2 - c^2| = |x - c||x + c| < \delta|x + c|$.

Since $|x +c| --> 2c$, and $|\delta| --> 0$ their product can be made as small as you want, and we are done.

I think c^4 case goes right off c^2, but I have no idea where to start with c^3. Also, is there a way to make the proof above a little more rigorous?

Thanks!

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# Homework Help: Another proof of limit by definition

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