When does a sequence $x(n)$ actually achieve the value at the limit?

Question

For example, $x(n) = 1/n$ has a limit of $0$ as $n \to \infty$ but does not achieve this limit, i.e., there does not exist a $N \in \mathbb{N}$ such that $x(n) = 0, \forall n \geq N$.

Under what assumption does this limit actually get achieved?

Answer 1

Assuming that eventually $x(n)=L$ we have $\exists N$, $\forall \varepsilon >0$, $\forall n\ge N$ and therefore

$$|x(n)-L|=0\le \epsilon \implies x(n) \to L$$

otherwise, if a constant value is not eventually achieved, limit might exist, as for your example $x(n)=1/n$ or not as for example $x(n)= \sin n$.