limites

Limits are a foundational concept in calculus and analysis. The limit of a function f as x approaches a is the value that f(x) approaches when x gets arbitrarily close to a (with x not necessarily equal to a). If such a value L exists, we write lim_{x→a} f(x) = L. If x tends to infinity, we speak of limits at infinity; if f(x) grows without bound, the limit is ±∞; if f oscillates without settling, the limit does not exist.

These ideas are made precise by the epsilon-delta definition: lim_{x→a} f(x) = L means that for every ε>0 there exists δ>0 such that 0<|x−a|<δ implies |f(x)−L|<ε. For sequences, a_n → L means that for every ε>0 there exists N such that n≥N implies |a_n−L|<ε. The two definitions are equivalent.

Examples and laws: Common limits include lim_{x→0} (sin x)/x = 1 and lim_{x→a} (f(x) + g(x)) = lim f(x) + lim g(x) when both limits exist. Limits respect addition, multiplication, and division by nonzero limits. Limits underpin continuity: a function is continuous at a if lim_{x→a} f(x) = f(a).

Applications and history: Limits are used to define derivatives, integrals, and power series, and to describe asymptotic behavior. The formal notion emerged in 19th-century analysis, with contributions from Cauchy and Weierstrass.