Limit

A limit describes the value that a function or sequence gets arbitrarily close to as its input approaches a specified point or becomes large. For a function f, the limit of f(x) as x approaches a is a number L such that f(x) can be made arbitrarily close to L by taking x sufficiently close to a (but not equal to a in the case of a finite limit). In rigorous terms, the epsilon-delta definition states that for every ε > 0 there exists δ > 0 with 0 < |x − a| < δ implying |f(x) − L| < ε. For a sequence a_n, the limit as n → ∞ is L if, for every ε > 0, there exists N such that n ≥ N implies |a_n − L| < ε.

Notation and types: lim_{x→a} f(x) = L. One-sided limits are lim_{x→a+} f(x) and lim_{x→a−} f(x). If the values grow without bound, we write lim_{x→a} f(x) = ∞ or −∞; when the limit does not exist in the finite or infinite sense, it is said to not exist. Limits at infinity are written lim_{x→∞} f(x) = L and lim_{x→−∞} f(x) = L.

Examples: lim_{x→0} (sin x)/x = 1. lim_{x→1} (x^2 − 1)/(x − 1) = lim_{x→1} (x + 1) = 2. lim_{x→0} 1/x does not exist as a finite number; it tends to ∞ from the right and −∞ from the left.

Limit laws: If lim f(x) = F and lim g(x) = G, then lim (f + g) = F + G, lim (fg) = FG, and lim (f/g) = F/G provided G ≠ 0 and the limits exist. If g(x) → a and f is continuous at a, then lim f(g(x)) = f(a). Limits underpin continuity, derivatives, and series.