limit (sequence)

We denote \[L\] as the limit of a sequence \[\left( x_{n} \right)\] if the numbers in the sequence becomes closer and closer to \[L\], e.g. \[x_{n}=\frac{1}{n}\], then \[x_{n}\to0\].

We call \[L\] the limit of the sequence \[\left( x_{n} \right)\], which is written as \[x_{n}\to L\] or \[\lim_{n\to\infty}x_{n}=L\] if the following holds: for each real \[\epsilon>0\], there exists a natural number \[N\] (the index) such that for every \[n\ge N\], \[\left| x_{n}-L \right|<\epsilon\].

Which essentially tells us that if \[\left( x_{n} \right)\] tends to \[L\], then we can always find an element in \[\left( x_{n} \right)\] such that its distance from \[L\] is less any given (very tiny) number, implying that the values in this sequence will end up very very close to \[L\], thus satisfying the epsilon-delta definition of limit.

The usual properties of limits also apply here, e.g. \[\lim_{n\to\infty}\left( a_{n}\pm b_{n} \right)=\lim_{a\to\infty}a_{n}\pm\lim_{n\to\infty}b_{n}\].

Similarly, for sequences that tend to infinity (or negative infinity), we write it as \[\lim_{n\to\infty}x_{n}=\pm\infty\] if the following holds: for each real number \[K\], there is a natural number \[N\] such that for every \[n\ge N\], \[x_{n}>K\] (or \[x_{n}<K\] for negative infinity); that is the terms in the sequence are eventually larger (or smaller) than any given real number, thus satisfying the concept of infinity.