Wrong Wolfram Alpha limit? $\lim\limits_{(x,y)\to(0,0)}\frac{{xy^3}}{{x^2+y^6}}$ Wolfram Alpha gives me: $\lim\limits_{(x,y)\to(0,0)} \dfrac{{xy^3}}{{x^2+y^6}}=0$.
I assume $f\left( {x,y} \right) = \dfrac{{x{y^3}}}{{{x^2} + {y^6}}}$
I find that when $x=y^3$, $f\left( {x,y} \right) = \dfrac{1}{2}$, while when $x=0, f\left( {x,y} \right) = 0$. Hence the limit does not exist.
Please correct me if I am wrong. Thank you.
 A: Mathematica, and consequently WolframAlpha, does not have built-in capability to evaluate arbitrary multivariate limits.  Therefore, the command
Limit[x y/(x^2 + y^2), x -> 0]

gives $0$, but
Limit[x y/(x^2 + y^2) /. x -> y, {y -> 0}]

yields $1/2$.  The path-dependence of the limit can only be handled when a path is specified.  This is something that many users of WolframAlpha misunderstand about Limit.  If you write
Limit[x y/(x^2 + y^2), {x -> 0, y -> 0}]

Mathematica will simply thread Limit[] over the list of replacement rules {x -> 0, y -> 0}, producing the output $\{0,0\}$.  To get a better understanding of what it's doing, try the command
Limit[x y/(x^2 + y^2), {x -> 1, y -> 2, x -> 3}]

Clearly, it doesn't behave the way one might think it should behave.  And if you think you can write
Limit[x y/(x^2 + y^2), {x,y} -> {0,0}]

you get an error, because {x,y} is not a valid variable for Limit.  This error definitely shows that such multivariate limits are not within the scope of Mathematica's functionality.

Now, that doesn't mean that WolframAlpha can't be a bit smarter than Mathematica's built in Limit:  if you enter
http://www.wolframalpha.com/input/?i=Limit+of+x+y%2F(x%5E2%2By%5E2)+as+(x,y)+tends+to+(0,0)
You will get the proper response, but if you enter
http://www.wolframalpha.com/input/?i=Limit+of+x+y%5E3%2F(x%5E2%2By%5E6)+as+(x,y)+tends+to+(0,0)
you get a wrong response, because again, WolframAlpha is built on Mathematica's functionality.  While it can handle some special queries, I suspect that these are probably pre-programmed somehow, or that they were special (but common) queries that were added to the knowledge base.  In any case, I would not rely on WolframAlpha or Mathematica to handle multivariate limits.
