WolframAlpha says limit exists when it doesn't? I was trying to calculate the following limit:

$$
\lim_{(x,y)\to (0,0)} \frac{(x^2+y^2)^2}{x^2+y^4}
$$

and, feeding it into WolframAlpha, I obtain the following answer, stating the limit is $0$:

However, when I try to calculate the limit when $x = 0$ and $y$ approaches 0, the limit is 1...
Is the answer given by WolframAlpha wrong? or am I?

This is only to complement the excellent answer of StackTD, who correctly shows that you are right — the limit does not exist (as one can find two different paths to the origin along which the limits of the function differ). The key message is:

Do not try limits in more than one variable with Mathematica or WolframAlpha.

(actually, I'd go further and suggest: Do not try limits in more than one real variable with Mathematica or WolframAlpha.)
See e.g. this thread on Mathematica.SE which goes at length into explaining one can possibly try and do it -- spoiler, it's complicated. Quoting a comment from there, by Jens:

With Limit, you're always restricted to a line in the larger space, and you can't make statements about the existence of the limit in the sense of the higher-dimensional space. For that you have to show the independence of the result on the direction of the line. If you intentionally set up a function to have different limits along different lines, I dont (sic) see what else you can do with Mathematica.

