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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.

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    $\begingroup$ You are correct, but how do you enter this into wolfram alpha? It could be good to provide a link. $\endgroup$ – mickep Dec 11 '16 at 7:30
  • $\begingroup$ (wolframalpha.com/input/…) It seems that the particular path along the x-axis is default? $\endgroup$ – Lee David Chung Lin Dec 11 '16 at 7:33
  • $\begingroup$ I also see "Standard computation time exceeded...", maybe this explains the wrong result. $\endgroup$ – user261263 Dec 11 '16 at 7:36
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    $\begingroup$ i think the limit does not exist $\endgroup$ – Dr. Sonnhard Graubner Dec 11 '16 at 7:55
  • $\begingroup$ @mickep This is the link I entered wolframalpha.com/input/… $\endgroup$ – dienhosp3 Dec 11 '16 at 7:58
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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.

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    $\begingroup$ On the other hand, Maple does attempt to compute actual multivariate limits. In this case, limit(x*y^3/(x^2+y^6),{x=0,y=0}); returns $undefined$. $\endgroup$ – Robert Israel Dec 12 '16 at 19:25

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