Definition:Limit of Sequence/Real Numbers

Definition
Let $\left \langle {x_n} \right \rangle$ be a sequence in $\R$.

Let $\left \langle {x_n} \right \rangle$ converge to a value $l \in \R$.

Then $l$ is a limit of $\left \langle {x_n} \right \rangle$ as $n$ tends to infinity.

This is usually written:
 * $\displaystyle l = \lim_{n \mathop \to \infty} x_n$

Also see

 * Convergent Real Sequence has Unique Limit

Also known as
A limit of $\left \langle {x_n} \right \rangle$ as $n$ tends to infinity can also be presented more tersely as a limit of $\left\langle {x_n} \right\rangle$ or even just limit of $x_n$.

Some sources present $\displaystyle \lim_{n \mathop \to \infty} x_n$ as $\lim_n x_n$.