I don't know how to type with the math code, so this may look weird but bear with me please.
(x,y) -> (0,0) x^2 + y^2
I have already figured that the limit does not exist, because it comes out to 0/0, however, I have to prove it through paths. I have one path where I changed the y to another x, and got the answer of 0, however, i have been going through multiple paths and can not get anything but 0 to compare it with. Any help is appreciated.