directionallimit

Directional limit is a concept in multivariable calculus describing the limit of a function as its input approaches a point along a fixed direction. Formally, let f: D ⊆ R^n → R and a be a point in the closure of D. For a nonzero vector u such that a + t u ∈ D for t near 0, the directional limit of f at a in the direction u is L = lim_{t→0} f(a + t u), if this limit exists. Often u is taken as a unit vector, but scaling does not change the existence or value of the limit.

Relation to full limits and continuity: If f is defined at a and continuous there, then for every direction u the limit of f(a + t u) as t → 0 equals f(a), so all directional limits exist and equal f(a). Conversely, if the full limit lim_{x→a} f(x) exists, then all directional limits exist and equal that same value. However, the existence of directional limits in every direction does not guarantee the full limit exists.

Example: Consider f(x,y) = (x^2 − y^2)/(x^2 + y^2) for (x,y) ≠ (0,0). The directional limit as (x,y) → (0,0) along the line y = m x is L(m) = lim_{t→0} f(t, m t) = (1 − m^2)/(1 + m^2), which depends on m. Therefore the full limit as (x,y) → (0,0) does not exist. Along y = 0 the limit is 1; along x = 0 the limit is −1.

Notes: The path must stay within the domain near a. Directional limits are a useful diagnostic for local behavior, but they do not alone determine continuity or differentiability at a.