Limit at origin of function of two variables

Question

How I can show that this limit does not exist at $(0,0)$ using some path. $f(x,y)=2xy/(x^{2}+y)$ where function has value $0$ at origin. Looking at graph of function,I got there's some parabola of type $y^{2}=-ax$, which will create some problem, but I am stuck.

Answer 1

Note that

• $x=0 \implies f(0,y)=0$
• $x=t \quad y=t^4-t^2 \quad t\to 0^+$ $$\implies \frac{2xy}{x^2+y}=\frac{2t^5-2t^3}{t^2+t^4-t^2}=\frac{2t^2-2}{t}\to-\infty$$