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I was looking to prove using the $\epsilon,\delta$ limit definition that $\lim_{x\to a}(\sqrt[3]{x})=\sqrt[3]{a}$, $(a>0)$. I'm not sure what sort of algebraic manipulation I should use on the expression $|\sqrt[3]{x}-\sqrt[3]{a}|$ (so I'll be able to continue with proving the limit).

Just a little hint would be great, thanks in advance.

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    Remember $a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b+...+ab^{n-2}+b^{n-1})$2012-11-09

3 Answers 3

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Multiply and divide $\sqrt[3]{x}-\sqrt[3]{a}$ by conjugate: $\dfrac{\left(\sqrt[3]{x}-\sqrt[3]{a} \right) \left(\sqrt[3]{x^2} +\sqrt[3]{x}\cdot\sqrt[3]{a} +\sqrt[3]{a^2} \right) }{\sqrt[3]{x^2} +\sqrt[3]{x}\cdot\sqrt[3]{a} +\sqrt[3]{a^2}}=\dfrac{x-a}{\sqrt[3]{x^2} +\sqrt[3]{x}\cdot\sqrt[3]{a} +\sqrt[3]{a^2}}$

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HINT: Use the identity $x^3-y^3=(x-y)(x^2+xy+y^2)$:

$\sqrt[3]x=\sqrt[3]a=\left(\sqrt[3]x-\sqrt[3]a\right)\cdot\frac{x^{2/3}+x^{1/3}a^{1/3}+a^{2/3}}{x^{2/3}+x^{1/3}a^{1/3}+a^{2/3}}$

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You can use the identity $ x - a = \left( \sqrt[3]{x} - \sqrt[3]{a} \right) \left( \sqrt[3]{x^{2}} + \sqrt[3]{x} \cdot \sqrt[3]{a} + \sqrt[3]{a^{2}} \right), $ which is derived from the following identity: $ a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2}). $ Then $ \forall x \in \mathbb{R} \setminus \{ a \}: \quad \sqrt[3]{x} - \sqrt[3]{a} = \frac{x - a}{\sqrt[3]{x^{2}} + \sqrt[3]{x} \cdot \sqrt[3]{a} + \sqrt[3]{a^{2}}}. $ Now, fix $ \epsilon > 0 $. Choose $ x \in \mathbb{R} $ so that $ |x - a| < \min \left( \frac{a}{2},\sqrt[3]{a^{2}} \cdot \epsilon \right). $ As $ a > 0 $, having $ |x - a| < \dfrac{a}{2} $ ensures that $ x > 0 $, and so $ \sqrt[3]{x^{2}} + \sqrt[3]{x} \cdot \sqrt[3]{a} + \sqrt[3]{a^{2}} > \sqrt[3]{a^{2}}. $ Next, having $ |x - a| < \sqrt[3]{a^{2}} \cdot \epsilon $ yields $ \left| \sqrt[3]{x} - \sqrt[3]{a} \right| < \epsilon. $ You can therefore set $ \delta := \min \left( \dfrac{a}{2},\sqrt[3]{a^{2}} \cdot \epsilon \right) $.

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    I would like to do that, but I want x > 0 so as to be able to exploit the inequality \sqrt[3]{x^{2}} + \sqrt[3]{x} \cdot \sqrt[3]{a} + \sqrt[3]{a^{2}} > \sqrt[3]{a^{2}} . If I choose $ \delta = \sqrt[3]{a^{2}} \cdot \epsilon $, then this $ \delta $ may be too big to be able to force $ x $ close enough to $ a $ so that it stays positive. Also, $ \epsilon $ is arbitrary here, so it can be as large as you please. You can try to see if there is a simpler $ \delta $ that takes care of everything though. I believe it is possible.2012-11-09