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
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
Notes: The path must stay within the domain near a. Directional limits are a useful diagnostic for