One-sided Limit

analysis

Definition

Definition of 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:

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

or:

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

with $x_n\in D$.

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Definition of 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 $x_0$ if:

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

or:

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

with $x_n\in D$.

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Implications of Equality

Let $f:D\to \mathbb{R}$ be a function with ${\lim }_{x\to x_0^{-}}f(x)={\lim }_{x\to x_0^{+}}$. This implies that the general limit at $x_0$ exists ("$⟹$").

If a general limit exists for $f$, this implies that ${\lim }_{x\to x_0^{-}}f(x)={\lim }_{x\to x_0^{+}}$ ("$⟸$").

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