In my calculus book it says that the limit of y(x^3) as (x,y)->(0,1) equals 0. It also says that a limit does not exist if you obtain different values when approaching (0,1) from different paths. It is easy to see the limit is zero by using the product rule for limits. However, if I set x=y, we get the limit of x^4 as (x,y)->(0,1) or the limit of y^4 as (x,y)->(0,1) which are clearly not equal. Hence the limit does not exist (?). Where is my reasoning false?