Function/R/Limit/Epsilon/Definition

Limit of a function

Let ${}T\subseteq \mathbb {R}$ denote a subset and ${}a\in \mathbb {R}$ a point. Let

$f\colon T\longrightarrow \mathbb {R}$

be a function. Then ${}b\in \mathbb {R}$ is called limit of ${}f$ in ${}a$, if for every ${}\epsilon >0$ there exists some ${}\delta >0$ such that for all ${}x\in T$ fulfilling

${}\vert {x-a}\vert \leq \delta \,,$

the estimate

${}\vert {f(x)-b}\vert \leq \epsilon \,$

holds. In this case, we write

${}\operatorname {lim} _{x\rightarrow a}\,f(x)=b\,.$