bf5dcf…: “Beyond a quiet solution”

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Consider the function $F(x,y)= \frac {2x^3y-2x^2y^2}{x^4+5y^4}$ .

As $\vector {x,y} \to \vector {0,0}$ along $x=0$ , $F(x,y)$ approaches $\answer {0}$ .
As $\vector {x,y} \to \vector {0,0}$ along $y=0$ , $F(x,y)$ approaches $\answer {0}$ .

Is this enough to determine that $\Lim {x,y}{0,0} = 0$ ? YesNo

As $\vector {x,y} \to \vector {0,0}$ along $y=x$ , $F(x,y)$ approaches $\answer {0}$ .

Is this enough to determine that $\Lim {x,y}{0,0} = 0$ ? YesNo

As $\vector {x,y} \to \vector {0,0}$ along $y=2x$ , $F(x,y)$ approaches $\answer {-\frac {4}{81}}$ .

So we conclude that:

$\Lim {x,y}{0,0} F(x,y)$ exists. $\Lim {x,y}{0,0} F(x,y)$ does not exist.

Take a minute to reflect on this; in order to determine that a limit exists, we have to show that the function tends to the same value along any path, not just check a few! If you check finitely many paths, you can never use the result to determine that a limit exists.