limx0

limx0 denotes the limit of a function as the variable x approaches zero, written as lim_{x→0} f(x). The concept focuses on the behavior of f near zero, not necessarily the value at x = 0. A limit may exist even if f(0) is undefined or different from the limiting value.

A limit can be finite, infinite, or nonexistent. The two-sided limit lim_{x→0} f(x) exists when both the left-hand limit lim_{x→0-} f(x) and the right-hand limit lim_{x→0+} f(x) exist and are equal. If they are not equal or either does not exist, the two-sided limit does not exist. When the function grows without bound as x approaches 0, we say the limit is infinite (e.g., lim_{x→0} 1/x^2 = ∞). Limits can also fail to exist due to oscillation or other irregular behavior (e.g., lim_{x→0} |x|/x does not exist).

Commonly used methods to evaluate lim_{x→0} f(x) include direct substitution when possible, algebraic rearrangement, the squeeze theorem, and standard limit results such as lim_{x→0} sin x / x = 1. Other tools include L’Hôpital’s rule for certain indeterminate forms and power series expansions for smooth functions.

Relation to continuity: a function is continuous at 0 if lim_{x→0} f(x) = f(0). Limits also underpin the definition of the derivative, via lim_{h→0} (f(0+h) − f(0))/h. In summary, lim_{x→0} f(x) captures the local behavior of f near zero and is a fundamental building block in analysis.