Limit of function with 3 variables

[halcyone]

Homework Statement

Evaluate the following limit as (x,y,z) --> (0,0,0), if it exists:

f(x,y,z)=xyz/x^3+y^3+z^3

2. Other relevant equations

I already know that f(x,y)=xy/x^3+y^3 doesn't exist, from a previous exercise. Hence, I suspect that this limit does not exist.

The Attempt at a Solution

I've taken the limit as we go along the x-axis, y=0, z=0, x-->0:

lim (x)-->0 0/x^3 = 0.

But, from the equation, you can tell that if you let any of x, y or z=0, you will get the limit to be equal to 0. I'm out of ideas on ways to approach this. I've tried letting y=x, but I can't cancel x or z out from the equation to make the limit not equal to 0.

I would appreciate any hints! Thanks.

 

[Dick]

Ok, it approaches zero along the x axis. Try approaching 0 from a different direction. Say, set x=y=z=a and let a->0.