# Another two variable limit

## Homework Statement

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

## 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?

Last edited:

Along x=y, the limit becomes

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

Thanks, I got it!

HallsofIvy