= 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 $\varepsilon_0>0$ permits a choice of $x_n$ with
$$
0<d_X(x_n,a)<1/n,\qquad d_Y(f(x_n),L)\geq\varepsilon_0.
$$
This <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>.
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