# find the limit of $\lim_{\ (x,y)\to (0,0) }\frac{xy^4}{(x^2+y^8)}$

This problem was on a recent quiz. I set $x$ and $y = 0$ and both limits approach $0$. Then I set $y=x$ and that limit also approaches $0$. I would think the answer is that it approaches $0$ but TA says that the limit does not exist. I plugged it into wolfram alpha and it also says limit is $0$.

So, does the limit exist? If so, what is it?

## 1 Answer

Take $x=y^4$, and the limit is not $0$ along this path.

• How would you know whether to set it at x=y^4 vs y=x? – user1766888 Feb 14 '13 at 20:41
• It is (usually) pretty easy to get $0$, so I try to figure out how to get non zero result. The numerator tends to zero no matter what we do to it, so the only way to "fix" it, is to cancel it (using the denominator)... And in the denominator we have $y^8$.... – Ludolila Feb 14 '13 at 20:47
• Ok. I'll remember that strategy next time before jumping to conclusions. Thanks. – user1766888 Feb 14 '13 at 20:53
• And take $x=y$ to see that the limit does not exist... – Julien Feb 14 '13 at 21:14