Sequential criterion for a function limit

For a function $f:D\to Y$ between metric spaces, and an accumulation point $a$ of $D\subseteq X$, the limit of a function satisfies ${\lim }_{x\to a}f(x)=L$ exactly when every sequence $x_n\in D\setminus \{a\}$ converging to $a$ has $f(x_n)\to L$.

The forward implication substitutes eventual closeness of $x_n$ into the epsilon-delta condition. For the converse, if that condition fails, some ${\epsilon }_0>0$ permits a choice of $x_n$ withThis sequence approaches $a$ while its images fail to approach $L$, a contradiction.

For $f(x)=\sin (1/x)$ near zero, take $x_n=(2\pi n+\pi /2{)}^{-1}$ and $y_n=(2\pi n+3\pi /2{)}^{-1}$. Both sequences tend to zero, but their images are $1$ and $-1$. Hence the limit of a function does not exist there. The metric-space hypothesis matters: sequences do not detect all limits in arbitrary topological spaces.