Left-sided Limit

analysis

Definition

Left-sided Limit

Let $f:D\to \mathbb{R}$ be a function.$f$ has a left-sided limit$c\in \mathbb{R}$ at $x_0\in D$ if the function values approach $c$ as $x$ approaches $x_0$ from the left ($x<x_0$).

$$\underset{x\to x_0^{-}}{\lim }f(x)=c⟺\forall \epsilon >0:\exists \delta >0:\forall x\in D:(x_0-\delta <x<x_0⟹|f(x)-c|<\epsilon )$$

Equivalently, using the sequential definition:

$$\underset{x\to x_0^{-}}{\lim }f(x)=c⟺\forall (x_n)\subseteq D:(x_n<x_0\wedge \underset{n\to \infty }{\lim }{x}_n=x_0⟹\underset{n\to \infty }{\lim }f(x_n)=c)$$