Limit (mathematics) - Limit of A Sequence

Limit of A Sequence

Consider the following sequence: 1.79, 1.799, 1.7999,... It can be observed that the numbers are "approaching" 1.8, the limit of the sequence.

Formally, suppose a₁, a₂, ... is a sequence of real numbers. It can be stated that the real number L is the limit of this sequence, namely:

to mean

For every real number ε > 0, there exists a natural number n₀ such that for all n > n₀, |a_n − L| < ε.

Intuitively, this means that eventually all elements of the sequence get arbitrarily close to the limit, since the absolute value |a_n − L| is the distance between a_n and L. Not every sequence has a limit; if it does, it is called convergent, and if it does not, it is divergent. One can show that a convergent sequence has only one limit.

The limit of a sequence and the limit of a function are closely related. On one hand, the limit as n goes to infinity of a sequence a(n) is simply the limit at infinity of a function defined on the natural numbers n. On the other hand, a limit L of a function f(x) as x goes to infinity, if it exists, is the same as the limit of any arbitrary sequence a_n that approaches L, and where a_n is never equal to L. Note that one such sequence would be L + 1/n.