# Proving a limit using epsilon/delta

1. Dec 5, 2009

### ImAnEngineer

1. The problem statement, all variables and given/known data
Prove that $$\mathop {\lim }\limits_{(x,y,z) \to (0,0,0) } \frac{xyz}{{x^2+y^2+z^2 }} = 0$$
using the epsilon-delta method

3. The attempt at a solution
$$0<|(x,y,z)-(0,0,0)|=\sqrt{x^2+y^2+z^2}<\delta$$
Now I have to rewrite:
$$0<\left|\frac{xyz}{{x^2+y^2+z^2 }}-0\right|<\epsilon$$
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:
$$0<\left|\frac{xyz}{{x^2+y^2+z^2 }}-0\right|\leq \left|\frac{xyz}{{x^2}}\right|=\left|\frac{yz}{{x}}\right|$$

Any help is very much appreciated!

Last edited: Dec 5, 2009
2. Dec 5, 2009

### grief

Hint: yz<=1/2*(y^2+z^2)

3. Dec 6, 2009

### ImAnEngineer

Then it is easy:

$$0<\left|\frac{xyz}{x^2+y^2+z^2}-0\right|\leq\left|\frac{xyz}{y^2+z^2}\right|\leq\left|\frac{xyz}{2yz}\right|=\left|\frac{x}{2}\right|<\frac{\delta}{2}=\epsilon$$

How did you come up with: yz<=1/2*(y^2+z^2) ?
(Why is it true at all?)