4
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Let's consider the limit

Limit[(x^2 + y^2)^2 Log[-(x^2 + y^2)], {x, y} -> {0, 0}]
(* 0 *)

Why if Mathematica takes the limit with a specific direction, eg.

Limit[(x^2 + y^2)^2 Log[-(x^2 + y^2)], {x, y} -> {0, 0}, Direction->-1]

doesn't give the same solution as before?

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  • $\begingroup$ Limit[(x^2 + y^2)^2 Log[-(x^2 + y^2)], Thread[{x, y} -> {0, 0}], Direction -> -1] does give 0. $\endgroup$ – kglr Oct 25 '17 at 7:58
  • $\begingroup$ yes, but the two limits, the one with {x, y} -> {0, 0} and the one with {x -> 0, y -> 0}, should coincide, also if, in general, they don't. $\endgroup$ – Giancarlo Oct 25 '17 at 8:13
  • $\begingroup$ Giancarlo, I agree. It seems that the form {x, y} -> {0, 0} is a new feature (it does not work in version 9) . It is possibly an oversight that {x, y} -> {0, 0} does not work when the option Direction is used. $\endgroup$ – kglr Oct 25 '17 at 8:26
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    $\begingroup$ I would report this to Wolfram Support. Limit got an overhaul recently. Multivariate limits are also a new feature (11.2 I think). It may still be a bit rough around the edges. $\endgroup$ – Szabolcs Oct 25 '17 at 8:42
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Up to http://reference.wolfram.com/language/ref/Limit.html , the correct syntax is

Limit[f[x, y], {x, y} -> {0, 0}, Direction -> {Reals, "FromBelow"}]
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  • $\begingroup$ If in your code I substitute f[x,y] with my function, it doesn't give a numbers as well. Moreover from reference.wolfram.com/language/ref/Direction «Approaching the origin from the first quadrant [...] Direction -> "FromAbove" [...] Equivalently: [...] Direction -> {"FromAbove", "FromAbove"}» $\endgroup$ – Giancarlo Oct 25 '17 at 8:24
  • $\begingroup$ That's a new syntax they added fairly recently, most likely because people found the +1/-1 specification confusing. -1 is actually "FromAbove". $\endgroup$ – Szabolcs Oct 25 '17 at 8:39
  • $\begingroup$ @user64494: In version 11.2, you syntax with option Direction -> {Reals, "FromBelow"} returns unevaluated (after defining f[x_, y_], of course). $\endgroup$ – murray Nov 24 '17 at 16:01

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