Evaluating the limit of a triple-variable function

[Elbobo]

Homework Statement

Find
$$\lim_ {(x,y,z) \rightarrow (0,0,0)} \frac{xy + 2yz^2 + 3xz^2}{x2 + y2 + z4}$$
if it exists.

The Attempt at a Solution

Not really sure how to evaluate limits with three variables, I tried letting x=y=0, then letting z approach 0. Then y=z=0, letting x approach 0. Then z=x=0, letting y approach 0. All three limits produced 0, but the answer is not 0.

 

[fzero]

You could try paths away from some of the axes, like z=0, x=y.

 

[Elbobo]

So I'm just supposed to do trial-and-error to find a path that proves the limit doesn't exist? How would I approach a problem where the limit does exist?

For the two-variable case, I would also use x=0 then y=mx, which would take care of all paths and leave no room for any misses. Is there a way to do such a thing in the three-variable case, at least at my level of math?

 

[fzero]

I think it is basically trial and error. Even using all lines may not be sufficient to show that a limit exists. For your function, you get different limits along the parabolae y=0, x=z² and x=0, y=z². By comparing relative powers of the variable in a rational function, you can usually determine the curves that you need to check.