# Direction in Limit

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?

• Limit[(x^2 + y^2)^2 Log[-(x^2 + y^2)], Thread[{x, y} -> {0, 0}], Direction -> -1] does give 0. – kglr Oct 25 '17 at 7:58
• 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. – Giancarlo Oct 25 '17 at 8:13
• 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. – kglr Oct 25 '17 at 8:26
• 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. – Szabolcs Oct 25 '17 at 8:42

Limit[f[x, y], {x, y} -> {0, 0}, Direction -> {Reals, "FromBelow"}]

• That's a new syntax they added fairly recently, most likely because people found the +1/-1 specification confusing. -1 is actually "FromAbove". – Szabolcs Oct 25 '17 at 8:39
• @user64494: In version 11.2, you syntax with option Direction -> {Reals, "FromBelow"} returns unevaluated (after defining f[x_, y_], of course). – murray Nov 24 '17 at 16:01