Function/R/Limit/Epsilon/Definition

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

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

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

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

the estimate

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

holds. In this case, we write

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