Limits

Limits are a fundamental concept in analysis that describe the value that a function or sequence approaches as its input or index approaches a given point or grows without bound. They formalize the idea of “approaching” and underpin derivatives, integrals, and continuity.

For a function f defined near a, the limit of f as x approaches a is L, written lim_{x->a} f(x) = L, if for every ε > 0 there exists δ > 0 such that 0 < |x - a| < δ implies |f(x) - L| < ε. This is the two-sided limit; one-sided limits lim_{x->a+} f(x) and lim_{x->a-} f(x) require x to approach a from the right or from the left. If no such L exists, the limit does not exist. Limits can also be infinite: lim_{x->a} f(x) = ∞ or -∞ describes unbounded growth as x approaches a, and lim_{x->∞} f(x) describes behavior as x grows without bound.

A limit of a sequence {a_n} as n approaches infinity is L if for every ε > 0 there exists N such that n ≥ N implies |a_n - L| < ε. This notion is called convergence.

Limit laws state that limits, when they exist, are preserved under addition, subtraction, multiplication, and division by a nonzero expression, and that composing with a continuous function preserves limits. Continuity at a point a means lim_{x->a} f(x) = f(a).

Indeterminate forms such as 0/0 or ∞/∞ require further analysis; techniques include algebraic simplification, series, or L’Hôpital’s rule. Examples include lim_{x->0} sin x / x = 1 and lim_{n->∞} (1 + 1/n)^n = e.