If we substitute x=0 and y=0 the result is 0/0 which is undefined. So we will arbitrarirly take y=kx.

Then:

(x²+3y²)/(x²-3y²)=(x²+3k²x²)/(x²-3k²x²)=x²(1+3k²)/x²(1-3k²)=(1+3k²)/(1-3k²)

The result depends on the value of k. For instance if k=2 result is -13/11, but if k=4 result is -49/47.

Therefore the limit does not exit.