Limit of a sequence

Given an infinite sequence $\{ x_n \}$ whose values are real numbers (or otherwise members of a common metric space), the limit of the sequence is a value $L$ which the values of the sequence approach, if such a value exists.

That is, we say $L=\lim_{n\to\infty}x_n$ , if for every $\varepsilon > 0$ , there exists a $N \in \N$ such that $k > N$ implies $|x_k-L|<\varepsilon$ (or $d(x_k,L)<\varepsilon$ in an arbitrary metric space).

That is, all values of the sequence after some $N$-th term are within $\varepsilon$ of the limit value. Put simply, this means that you can choose any "maximum distance" from an element of the sequence to its limit, and there will only be a finite number of elements that do not fall into this range, no matter how small the distance chosen.

The most common method of evaluating the limit of a sequence is to make a function $f(n)=a_n$ , and taking the limit at infinity. For example,

$\begin{align}&a_n=\frac{3n^2-5}{2n^2+n}\\ &\lim_{n\to\infty}a_n=\lim_{n\to\infty}f(n)=\lim_{n\to\infty}\frac{3n^2-5}{2n^2+n}=\frac32\end{align}$