Limit of a function at a point

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, let $E\subseteq X$ , let $x_0\in X$ be a limit point of $E$ (meaning every neighborhood of $x_0$ meets $E\setminus\{x_0\}$ ), and let $f:E\to Y$ . We say that $f(x)$ tends to $L\in Y$ as $x\to x_0$ , written

\lim_{x\to x_0} f(x)=L,

\forall \varepsilon>0,\ \exists \delta>0\ \text{such that}\ \forall x\in E,\ \bigl(0<d_X(x,x_0)<\delta \Rightarrow d_Y(f(x),L)<\varepsilon\bigr).

Examples

For $f:\mathbb{R}\setminus\{0\}\to\mathbb{R}$ given by $f(x)=\sin x/x$ , one has $\lim_{x\to 0} f(x)=1$ (a standard analytic limit).
For $f(x)=x^2$ on $\mathbb{R}$ , $\lim_{x\to a} f(x)=a^2$ for every $a\in\mathbb{R}$ .
If $f(x)=1/x$ on $\mathbb{R}\setminus\{0\}$ , then $\lim_{x\to 0} f(x)$ does not exist in $\mathbb{R}$ .

Remarks

This definition captures the local behavior of $f$ near $x_0$ independent of the value of $f$ at $x_0$ . It is the foundation for continuity and differentiation.