Answer
The limit does not exist
Work Step by Step
Consider $f(x,y)=\dfrac{xy^2-1}{y-1}$
Let us consider the first approach: $(x,y) \to (1,1)$ along $x=1$
Then, we get $\lim\limits_{y \to 1}\dfrac{y^2-1}{y-1}=\lim\limits_{y \to 1}\dfrac{(y-1)(y+1)}{y-1}=2$
Let us consider the second approach: $(x,y) \to (1,1)$ along $x=y$
Then, we get $\lim\limits_{y \to 1}\dfrac{y^3-1}{y-1}=\lim\limits_{y \to 1}\dfrac{(y-1)(y^2+y+1)}{(y-1)}=\lim\limits_{y \to 1} (y^2+y+1)=(1)^2+1+1=3$
This shows that there are different limit values when the approach is different and therefore, the limit does not exist for the given function.
