real analysis - Check the existence of partial derivatives and continuity of $f(x,y)$ at $(1,0)$ : $f(x,y) = frac{3y(x-1)} {{({x-1})^2} +y^2}$

Check the existence of partial derivatives and continuity of $f(x,y)$ at $(1,0)$ : $f(x,y) = \frac{3y(x-1)} {{({x-1})^2} +y^2}$ when $(x,y)\neq (1,0)$ and $0$ otherwise. I decided to check continuity first - I calculated limit by considering different paths of approach and I got $0$ in all cases (confirmed continuity). I checked that in Wolfram and found out that limit shouldn't exist. Where have I done mistake and how to check the existence of partial derivatives?

Answer

Note that if $x=y+1$, then

$$f(x,y)=\frac{3y^2}{2y^2}=\frac32$$ and that therefore if you approach $(1,0)$ along the line $x=y+1$, then the limit is $\frac32\neq0$.