Multivariate limit: more details following post

Question

anonymous · Answer

$$\lim_{(x,y) \rightarrow (0,0)} \frac{ 2xy }{ x²+y² }$$

I got a tip to solve it with y=kx

$$\lim_{x \rightarrow 0}\frac{ 2kx² }{ x²+kx }$$
Applying L'Hopital's Rules that becomes
$$\lim_{x \rightarrow 0}\frac{ 4kx }{ 2x+k } = \frac{ 0 }{ k }=0$$

Then I tried x=y²

$$\lim_{y \rightarrow 0} \frac{ 2y³ }{ 2y² } = \lim_{y \rightarrow 0} y = 0$$

Are there any other directions I can think of and not have it zero?

dumbcow · Answer

you missed something with your substitution of y=kx

you forgot to square it on bottom ....y^2 = k^2x^2

dumbcow · Answer

can the answer be in terms of k?

dumbcow · Answer

http://www.wolframalpha.com/input/?i=lim+2xy%2F%28x%5E2%2By%5E2%29+as+%28x%2Cy%29-%3E%280%2C0%29 seems we need more info has to how y and x are related

anonymous · Answer

for multivariate you can choose whatever between y and x. But you're right I made a mistake in the denominator, and get 1/k. And when I approach it from y, I get zero, which proves that the limit does not exist.

dumbcow · Answer

ahh ok