For the x=a path, the limit doesn't exist. Does the multivariable limit exist?

Question

I have the following limit:$$L=\lim_{(x,y) \to (1,1)} \dfrac{x^2+y^2-2}{|x-1|+|y-1|}$$ If I take the limit for $x=1$, then we get:$$L_1=\lim_{y \to 1_{+}}\dfrac{y^2-1}{|y-1|}=2$$, and $$L_2=\lim_{y \to 1_{-}}\dfrac{y^2-1}{|y-1|}=-2$$ So, because $L_1 \neq L_2$ does this mean that $L$ does not exist? If not how to solve the limit. I cannot use polar coordinates, nor the path $x=y$ (I get a similar result from the one cited above)