- #1

ImAnEngineer

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## Homework Statement

Prove that [tex]\mathop {\lim }\limits_{(x,y,z) \to (0,0,0) } \frac{xyz}{{x^2+y^2+z^2 }} = 0[/tex]

using the epsilon-delta method

## The Attempt at a Solution

[tex]0<|(x,y,z)-(0,0,0)|=\sqrt{x^2+y^2+z^2}<\delta[/tex]

Now I have to rewrite:

[tex]0<\left|\frac{xyz}{{x^2+y^2+z^2 }}-0\right|<\epsilon [/tex]

So that I find a relationship between epsilon and delta.

This is where I get stuck... I can't figure out how to do that.

This is one of my attempts:

[tex]0<\left|\frac{xyz}{{x^2+y^2+z^2 }}-0\right|\leq \left|\frac{xyz}{{x^2}}\right|=\left|\frac{yz}{{x}}\right| [/tex]

Any help is very much appreciated!

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