Show that the indicated limit does not exist for: lim┬((x,y)→(0,0))⁡〖(x^2+y^2)/(x^2+xy+y^2 )〗

Fix m, and let y = mx:

lim(x→0) (x^2 + (mx)^2) / (x^2 + x * mx + (mx)^2)
= lim(x→0) x^2 (1 + m^2) / [x^2 (1 + m + m^2)]
= (1 + m^2)/(1 + m + m^2).

Since this dependent on the value of m, the multivariable limit does not exist.

I hope this helps!