What is the limit of [2x^{2}y/(x^4 + y^4)] as (x,y) approaches (0,0)?

[merced]

Homework Statement

Find the limit, if it exists, or show that the limit does not exist.
$$lim_{(x,y)->(0,0)}[2x^{2}y/(x^4 + y^4)]$$

Homework Equations

The Attempt at a Solution

Along the y-axis and the x-axis, the limit approaches 0. Along y = mx, the limit also appaches 0. So, it appears that the limit is 0. However, the answer is that the limit "does not exist."

Should I just keep making new equations until I find where the limit does not = 0? I even tried the Squeeze Theorem...

0<$$[2x^{2}y/(x^4 + y^4)]$$<$$2x^2$$
because $$y/(x^4 + y^4)$$<1
so as x -> 0, the whole function -> 0 right?

Why doesn't that work to prove that the limit would be 0?

 

[IMDerek]

Along x=y, the limit becomes

$$\lim _{x \to 0} \frac {2x^3}{2x^4} = \lim _{x \to 0} \frac 1x$$

 

[merced]

Thanks, I got it!

 

[HallsofIvy]

Did you read my response to your first question? No matter how many curves you try you can never prove that a limit exists that way. In fact, it is possible to show that the limit is the same for all straight lines through the origin, that would still not show that the limit is the same for curved lines.