DoppelGrenzwerts

DoppelGrenzwerts is a concept in mathematical analysis describing a pair of limit values associated with a function of two variables as each variable approaches its respective limit. It formalizes the idea that the behavior of a function near a point (a,b) can be captured by two marginal limits, even when a single joint limit does not exist.

Formally, let f be a function of two variables defined on a domain D, and suppose (a,b) lies in D with a and b as limit points. The DoppelGrenzwerts of f at (a,b) is the ordered pair (Lx, Ly) where Lx = lim_{x→a} f(x,b) provided this limit exists, and Ly = lim_{y→b} f(a,y) provided this limit exists. If both limits exist, the pair (Lx, Ly) is the DoppelGrenzwerts. If a joint limit lim_{(x,y)→(a,b)} f(x,y) exists, it equals a common value L and then Lx = Ly = L.

Example: consider f(x,y) = x^2/(x^2 + y^2) near (0,0). For fixed y ≠ 0, lim_{x→0} f(x,y) = 0, so Lx = 0. For x = 0, lim_{y→0} f(0,y) = 1, so Ly = 1. The DoppelGrenzwerts at (0,0) are (0, 1), while the joint limit does not exist.

Origins and usage: the term is used primarily in theoretical discussions and some German-language expositions or speculative texts to discuss limit processes along coordinate directions. It is not a standard term in mainstream analysis, and definitions may vary between authors.

See also: limit, joint limit, path dependence, Fubini-type considerations.