directionallimiting

Directional limiting, or directional limit, refers to the limiting behavior of a function as its input approaches a point along a specified direction. Formally, for a function f: R^n → R (or C), a point a ∈ R^n, and a nonzero direction vector u, the directional limit of f at a in the direction of u is lim_{t→0} f(a + t u), if this limit exists. Some treatments use the one-sided limit t → 0+ (approaching along the ray a + t u with t ≥ 0), while others allow t to approach 0 from both sides.

Relation to the ordinary limit and continuity: If the full limit lim_{x→a} f(x) exists and equals L, then the directional limit along every direction u also exists and equals L. Conversely, if the directional limit exists for every direction u and yields the same value L, then the full limit lim_{x→a} f(x) exists and equals L. If directional limits depend on the direction, the full limit does not exist. Directional limits can therefore reveal whether a function is continuous at a point or whether its limit is path-dependent.

Examples: Consider f(x, y) = (x^2 − y^2)/(x^2 + y^2) near (0, 0). Along lines y = mx, the limit is (1 − m^2)/(1 + m^2), which varies with m, so the directional limit exists for each direction but not as a single value; thus the limit at the origin does not exist. In contrast, f(x, y) = (x y)/(x^2 + y^2) has direction-dependent limits as well, illustrating non-existence of a unique limit.

Notes: Directional limits are related to directional derivatives and are useful in analysis of differentiability and in characterizing continuity. See also limits, directional derivatives, Gateaux derivative, and Fréchet derivative.