Please help me: Find lim(x,y)->(0,0): xy/x+y

What I've done is:
Suppose xy>0
(|x| - |y|)^2=x^2+y^2-2|xy|
=> |xy| =< x^2 + y^2 .................... that's .1
Then we know that sqrt(x^2 + y^2) =< x + y
=> lim(x,y)->(0,0): xy/x+y =< lim(x,y)->(0,0): (x^2 + y^2)/sqrt(x^2 + y^2) = lim(x,y)->(0,0): sqrt(x^2 + y^2) and here the limit doesn't exist because if y=x
lim(x,y)->(0,0): sqrt(x^2 + y^2) = lim(x,y)->(0,0): sqrt(2x^2)
and if y = 0
lim(x,y)->(0,0): sqrt(x^2 + y^2) = lim(x,y)->(0,0): sqrt(x^2)

Did I make any mistake? If I did please give me the correct answer.
Thank you

This limit does not exist, since we get different limits on different paths to (0, 0).

Letting y = x and x→0, we have lim(x→0) x^2/(2x) = 0
Letting y = -x and x→0, we have lim(x→0) -x^2/0, which is infinite.

I hope this helps!

You have complicated things, because the answer is a lot simpler than that.

L = lim[(x, y) → (0, 0)] (xy / x + y)
L = lim[(x, y) → (0, 0)] (y + y)
L = lim[(x, y) → (0, 0)] 2y
L = 0