finding a function with 2 variables that has a limit equal to zero

Question

I am trying to find a function with $2$ variables that have a limit equal to zero at the point $(0,0)$ if you check the limit by any straight line that gets to that point, but also the function doesn't have a limit at the point $(0,0)$.

Answer 1

Consider: $$f(x,y)=\frac{yx^2}{y^2+x^4}$$ Verify that $f$ tends to $0$ along any straight line and try along the parabola $y=x^2$ as well.