The Definition of Limit

${\displaystyle ϵ-\delta }$ Definition

For a given ${\displaystyle ϵ>0}$, no matter how small it is, when ${\displaystyle |f(x)-L|<ϵ}$, there is always a range of ${\displaystyle 2\delta }$ surround ${\displaystyle c}$ (${\displaystyle 0<|x-c|<\delta }$) that makes the function exists, even ${\displaystyle f(c)}$ does not exists.

We say that in this case, ${\displaystyle \underset{x\to c}{lim}f(x)=L}$.