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,
Examples: Consider f(x, y) = (x^2 − y^2)/(x^2 + y^2) near (0, 0). Along lines y = mx, the limit
Notes: Directional limits are related to directional derivatives and are useful in analysis of differentiability and