The left and right limits of $f$ at $c$, as limits of the restrictions of $f$ to $A\cap (-\mathrm{\infty },c)$ and $A\cap (c,\mathrm{\infty })$

Let $A\subseteq \mathbb{R}$, let $f:A\to \mathbb{R}$ and let $c\in \mathbb{R}$. Put

$A^{-}:=A\cap (-\mathrm{\infty },c),A^{+}:=A\cap (c,\mathrm{\infty })$

(Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length), and write $f^{-}:=f{\mathrm{\mid }}_{A^{-}}$ and $f^{+}:=f{\mathrm{\mid }}_{A^{+}}$ for the restrictions of $f$ to those sets.

Right limit. Suppose $c$ is a limit point of $A^{+}$ (Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$). For $L\in \mathbb{R}$ we write

${\lim }_{x\to c^{+}}f(x)=L:⟺{\lim }_{x\to c}{f}^{+}(x)=L$

in the sense of The $\epsilon$-$\delta$ limit ${\lim }_{x\to c}f(x)=L$ of $f:A\to \mathbb{R}$ at a limit point $c$ of $A$. Written out: for every real $\epsilon >0$ there is a real $\delta >0$ such that

$\mathrm{\mid }f(x)-L\mathrm{\mid }<\epsilon \text{for every }x\in A\text{ with }c<x<c+\delta \mathrm{.}$

Left limit. Suppose $c$ is a limit point of $A^{-}$. For $L\in \mathbb{R}$ we write ${\lim }_{x\to c^{-}}f(x)=L$ when ${\lim }_{x\to c}{f}^{-}(x)=L$; written out, for every real $\epsilon >0$ there is a real $\delta >0$ with $\mathrm{\mid }f(x)-L\mathrm{\mid }<\epsilon$ for every $x\in A$ with $c-\delta <x<c$.

The written-out forms agree with the definitions. For $x\in A^{+}$ the two conditions $0<\mathrm{\mid }x-c\mathrm{\mid }<\delta$ and $c<x<c+\delta$ are the same: $x>c$ gives $x-c>0$, so $\mathrm{\mid }x-c\mathrm{\mid }=x-c$ and $0<\mathrm{\mid }x-c\mathrm{\mid }<\delta$ reads $0<x-c<\delta$ (Basic properties of the absolute value). Symmetrically on the left, where $x<c$ gives $\mathrm{\mid }x-c\mathrm{\mid }=c-x$.

Well-posedness is inherited, not reproved. A one-sided limit is a limit, namely the limit of a restriction, so:

• Uniqueness. At most one $L$ can occur, by At a limit point of the domain a function has at most one limit applied to $f^{+}$ on the domain $A^{+}$ (respectively to $f^{-}$ on $A^{-}$), which is legitimate exactly because $c$ was required to be a limit point of that set. This is what makes the notation ${\lim }_{x\to c^{+}}f(x)$ denote a single real.
• Locality and restriction. Both claims of The limit at $c$ depends only on the restriction of $f$ to a punctured neighbourhood of $c$, and passes to any subset of the domain having $c$ as a limit point apply verbatim to $f^{+}$ and $f^{-}$.

When the symbols are defined. If $c$ is not a limit point of $A^{+}$ — for instance if $A$ contains no point to the right of $c$, or only points bounded away from $c$ on that side — then ${\lim }_{x\to c^{+}}f(x)$ is not defined here, for the reason given in The $\epsilon$-$\delta$ limit ${\lim }_{x\to c}f(x)=L$ of $f:A\to \mathbb{R}$ at a limit point $c$ of $A$: the $\epsilon$-$\delta$ condition would be satisfied vacuously by every real at once. The same applies on the left.

• Neither one-sided limit requires $c\in A$, and neither looks at $f(c)$. Both properties are inherited from The $\epsilon$-$\delta$ limit ${\lim }_{x\to c}f(x)=L$ of $f:A\to \mathbb{R}$ at a limit point $c$ of $A$, since $c\mathrm{\notin }{A}^{-}\cup A^{+}$: the point $c$ belongs to neither $(-\mathrm{\infty },c)$ nor $(c,\mathrm{\infty })$.

• The two one-sided limits and the two-sided limit. When $c$ is a limit point of both $A^{-}$ and $A^{+}$, the two-sided limit exists exactly when both one-sided limits exist and agree, and then all three coincide: If $c$ is a limit point of the domain from both sides, the limit exists iff both one-sided limits exist and agree. When $c$ is a limit point of only one of the two sets, that one-sided limit and the two-sided limit are the same condition, again by claim 2 of The limit at $c$ depends only on the restriction of $f$ to a punctured neighbourhood of $c$, and passes to any subset of the domain having $c$ as a limit point together with the observation that $A$ and that one side have the same points in a small enough punctured neighbourhood of $c$.

• Notation. Some texts write $f(c+)$ and $f(c-)$ for these values. This library writes only ${\lim }_{x\to c^{+}}f(x)$ and ${\lim }_{x\to c^{-}}f(x)$, because the shorter notation looks like an evaluation of $f$ and these quantities are not values of $f$: they are defined without reference to $f(c)$, which may not even exist.