From this question on answers.yahoo, the guy says the following limit does not exist:
$$\lim_{(x,y) \to (0,0)} \frac{xy^4}{x^2 + y^8},$$
then on wolfram, it says the limit is equal to $0$. When I did it myself, I tried approaching $(0,0)$ from the $x$-axis, $y$-axis, $y=x$, $y=x^2$. They all equal $0$.
But when I tried the squeeze theorem, I got $y^8 \leq x^2 + y^8$, therefore $0 \leq |xy^4/(x^2+y^8)| \leq |\dfrac{x}{y^4}|$, and the latter does not exist for $(x, y) \to (0,0)$.
So does the original limit exist or not? I'm getting contradicting information from various sources. Also, if it doesn't exist (it looks like it doesn't… I think), how would I go about proving that it doesn't?
Best Answer
Approaching $(0,0)$ along the curve $x=y^4$ you get
$$ \frac{xy^4}{x^2+y^8} = \frac{y^8}{y^8 + y^8} = \frac12 $$
which clearly does not tend to $0$ as $y \to 0$. Since you get different values along different paths, the limit doesn't exist.