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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} $
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