limx

Limx is a fundamental concept in calculus and analysis describing the value that a function f(x) approaches as the variable x approaches a given point a. The standard notation is lim_{x→a} f(x). If the limit exists and equals L, we write lim_{x→a} f(x) = L; otherwise the limit does not exist or is said to diverge.

Limits can be taken from both sides or from one side only. One-sided limits are written as lim_{x→a^-} f(x) (approaching from the left) and lim_{x→a^+} f(x) (approaching from the right). Limits at infinity are written as lim_{x→∞} f(x) or lim_{x→-∞} f(x). The value of a limit is determined by the behavior of f(x) as x gets arbitrarily close to a, not by the value of f at a itself.

Existence of a limit depends on whether f(x) can be made arbitrarily close to a single number L by taking x sufficiently close to a (excluding a). Limits obey a set of algebraic laws: the limit of a sum is the sum of the limits, the limit of a product is the product of the limits, and so on, provided the individual limits exist. If f is continuous at a, then lim_{x→a} f(x) = f(a).

Common techniques include direct substitution when f is continuous, algebraic manipulation, factoring, rationalizing, and, in indeterminate cases, L'Hôpital's rule or the squeeze theorem. Limits are foundational for defining derivatives, integrals, and notions of convergence in analysis.