Consider the limit. Please show work why.

Consider the limit: lim (x,y) --> (0,0): (6xy)/(x^2+y^2) Please show how you obtained the answer.

Is it?:

1) The limit is 0
2) The limit is 3
3) The limit is all values between -3 and 3
4) The limit does not exist

Consider the limit along the path y = x. Then:
lim (x, y)-->(0, 0) 6xy/(x^2 + y^2)
= lim (x-->0) 6x^2/(x^2 + x^2)
= lim (x-->0) 6x^2/(2x^2)
= 3.

Then, consider the path along y = x^2.
lim (x, y)-->(0, 0) 6xy/(x^2 + y^2)
= lim (x-->0) 6x^3/(x^2 + x^4)
= lim (x-->0) 6x/(1 + x^2)
= 0.

Since the value of the limit differs depending on the path taken, the answer is (4).

I hope this helps!