Evaluate $\lim_{(x,y,z)\to (0,0,0)}\frac{xyz}{x^4+y^4+z^4}$

Question

Evaluate the limit if it exists $$\lim_{(x,y,z)\to (0,0,0)}\frac{xyz}{x^4+y^4+z^4}.$$

If we take the path $x=y=z$ we get

$$\lim_{(x,x,x)\to (0,0,0)}\frac{x^3}{3x^4}=\lim_{x\to 0}\frac{1}{3x}=\infty$$

Is it sufficient to show that the limit does not exist?

Answer 1

You should take another path as well, say x=y=-z and show the answer would be -infinity, that should the limit does not exist.

Answer 2

Actually if $\lim_{(x,y,z)\rightarrow(0,0,0)}f(x,y,z)=L$ exists as a real number, then $\lim_{x\rightarrow 0}f(x,x,x)$ exists as a real number and the limit is $L$, so your reasoning is solid in claiming that $\lim_{(x,y,z)\rightarrow(0,0,0)}f(x,y,z)$ does not exist as a real number. If one wants to claim that it does not exist in extended real numbers sense, one realizes to another path $x=y=-z$ as @anishtain4 has noted to get $-\infty$.