Can L'Hôpital's rule apply along a path in multivariable limits?

[Feodalherren]

Homework Statement

The limit as x,y → 0

$\frac{y^{2}Sin^{2}x}{x^{4}+y^{4}}$

Homework Equations

The Attempt at a Solution

There are pretty straight forward but I have a general question about them. So say in the function above I let y=x and let x approach 0.

I get 0/0 - inconclusive. Can I fall back on single variable calculus and use L'Hospital's rule to find the limit? For some reason my gut tells me no.

 

[STEMucator]

If you approach along a straight line, you would need to use L'Hospital's rule to find the limit. What happens if you approach along a parabola? Compare that to approaching along a straight line. What can you conclude?