Getting Closer and Closer: The Limit of a Sequence

A sequence is a function whose domain is the set of natural numbers (or a subset of integers starting from some integer). We often denote a sequence as {a_n}, where a_n is the n-th term of the sequence.

A sequence {a_n} is said to converge to a limit L if, for every positive number ε (epsilon), there exists a natural number N such that for all n > N, the absolute difference |a_n - L| < ε. This means the terms of the sequence eventually fall within any arbitrarily small interval around L.

The concept of a limit is the bedrock of calculus and real analysis. It allows for the rigorous definition of continuity, derivatives, and integrals. Understanding limits enables the study of infinite series and the behavior of functions as they approach specific points or infinity.

Determining the limit of a sequence often involves algebraic manipulation, applying limit laws (sum, product, quotient rules), using the Squeeze Theorem, or employing L'Hôpital's Rule for sequences that can be represented as functions of a real variable. Proofs often rely on the formal epsilon-N definition.

In physics, limits describe phenomena like the approach to equilibrium or terminal velocity. In computer science, they relate to the efficiency of algorithms. For instance, the sequence defined by a_n = 1/n converges to 0 as n approaches infinity. The sequence b_n = (n+1)/n converges to 1.