límite

Límite is a Spanish noun that denotes a boundary, end, or cap—the furthest extent or maximum that something can reach. It is used in everyday language to express constraints, borders, or thresholds. Etymology traces to Latin limitis, passing through Old French, and in Spanish it forms compounds such as límite superior (upper limit) and límite inferior (lower limit).

In mathematics, límite refers to the value that a function, sequence, or process approaches as its input or index tends toward a particular value. Notation commonly used is lim_{x→c} f(x) = L, with one-sided limits written as lim_{x→c+} f(x) and lim_{x→c−} f(x). If the value grows without bound or the input tends to infinity, we speak of infinite limits, e.g., lim_{x→c} f(x) = ∞ or lim_{x→∞} f(x) = L. A limit exists when the function approaches the same value from all applicable directions; otherwise, the limit does not exist. A formal epsilon-delta definition describes this precision: for every ε>0 there exists δ>0 such that 0<|x−c|<δ implies |f(x)−L|<ε. Limits underpin derivatives and integrals and give rise to many standard results, such as lim_{x→0} (sin x)/x = 1 and lim_{n→∞} 1/n = 0.

Beyond mathematics, límite encompasses practical boundaries: the maximum quantities permitted, borders of a region, or temporal constraints like a time limit. In daily life and applied fields, it appears in phrases such as speed limit, credit limit, and legal or regulatory caps, reflecting the idea of a defined confinement or stopping point.

In science and philosophy, limits conceptually describe universal or theoretical constraints, such as maximum speed in physics or epistemic boundaries in inquiry.