liminfxx0

Liminfxx0, in mathematics, refers to the limit inferior of a real-valued function at a point x0, commonly written as liminf_{x→x0} f(x). It measures the greatest lower bound of values that f(x) approaches as x tends to x0.

One standard definition uses shrinking neighborhoods: if f is defined in a neighborhood of x0, liminf_{x→x0} f(x) is the limit as r approaches 0 from above of the infimum of f on the punctured interval around x0, that is, lim_{r→0+} inf{ f(x) : 0<|x−x0|<r }. Equivalently, it can be described via sequences: liminf_{x→x0} f(x) equals the infimum of the set of all subsequential limits of f(xn) taken along sequences xn → x0 with xn ≠ x0, whenever these subsequences converge.

The liminf and limsup are related concepts: liminf_{x→x0} f(x) ≤ limsup_{x→x0} f(x). If f is continuous at x0, both values coincide and equal f(x0). The liminf is central to the notion of lower semicontinuity: f is lower semicontinuous at x0 if liminf_{x→x0} f(x) ≥ f(x0).

Example: if f(x) = sin(1/x) for x ≠ 0 and f(0) is defined arbitrarily, then liminf_{x→0} f(x) = −1 and limsup_{x→0} f(x) = 1, reflecting the oscillatory behavior near x = 0.

In domains extending to higher dimensions, liminf_{x→x0} f(x) is defined similarly using neighborhoods around x0, and it plays a key role in analysis, optimization, and variational methods.