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!
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!