limx2

limx2 refers to the limit of the square function f(x) = x^2 as x approaches a real number a. It is commonly written as lim_{x→a} x^2, and the central result is that for every finite real a, the limit equals a^2. This follows from the continuity of the squaring function on the entire real line, so the limit from the left and the right exists and equals the function value at a.

If a is infinite, the limit behaves differently: lim_{x→∞} x^2 = ∞ and lim_{x→−∞} x^2 = ∞, meaning the expression diverges in the real-number sense. The function x^2 is even, growing without bound as |x| increases, which underpins these divergent limits at infinity.

Examples illustrate the concept: lim_{x→3} x^2 = 9, lim_{x→−2} x^2 = 4, and lim_{x→0} x^2 = 0. These reflect the general principle that the limit at a finite point matches the value of the function there, due to continuity.

Properties and generalizations: Because x^2 is a polynomial, it is continuous for all real x, so lim_{x→a} x^2 = a^2 for any real a. More generally, for any polynomial p(x), lim_{x→a} p(x) = p(a). The result also extends to sequences: if x_n → a, then x_n^2 → a^2.

See also: limit, continuity, polynomial function, limit laws.