Limit Existence and the Epsilon-Delta Proof

[faradayscat]

Homework Statement

Prove that

lim (x,y,z)→(0,0,0) 2xz/(x²+y²+z²) = 0

Homework Equations

My teacher wants me to show this using epsilon delta, so

0<√(x²+y²+z²)<∂ ⇒ |f(x,y,z) - 0| < ε

The Attempt at a Solution

The limit does not exist apparently.. when you approach the limit along different paths you get different answers.. say, along (t,0,t) gives 1 while along (0,0,t) gives 0. Did my professor make a mistake in this assignment, or does the limit actually exist and I'm missing something?

 

[vela]

You're right. Your professor made a mistake.