One-sided Limit

analysis

Definition

A one-sided limit is the limit of a function as the input approaches a specific value from either the left or the right.

Left-sided Limit

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:

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

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Right-sided Limit

Definition

Right-sided Limit

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

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

Equivalently, using the sequential definition:

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

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Implications of Equality

A function $f$ has a two-sided limit at $x_0$ if and only if both one-sided limits exist and are equal:

$$\underset{x\to x_0}{\lim }f(x)=c⟺\underset{x\to x_0^{-}}{\lim }f(x)=\underset{x\to x_0^{+}}{\lim }f(x)=c$$

If the one-sided limits exist but are not equal, the function has a jump discontinuity at $x_0$.