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

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

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

(Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length), and write $f^{-} := f|_{A^{-}}$ and $f^{+} := f|_{A^{+}}$ 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

$\lim_{x \to c^{+}} f(x) = L \quad :\Longleftrightarrow \quad \lim_{x \to c} f^{+}(x) = L$

in the sense of The $\varepsilon$-$\delta$ 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 $\varepsilon > 0$ there is a real $\delta > 0$ such that

$|f(x) - L| < \varepsilon \qquad \text{for every } x \in A \text{ with } c < x < c + \delta .$

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 $\varepsilon > 0$ there is a real $\delta > 0$ with $|f(x) - L| < \varepsilon$ 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 $0 < |x - c| < \delta$ and $c < x < c + \delta$ are the same: $x > c$ gives $x - c > 0$, so $|x - c| = x - c$ and $0 < |x - c| < \delta$ reads $0 < x - c < \delta$ (Basic properties of the absolute value). Symmetrically on the left, where $x < c$ gives $|x - c| = c - x$.

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 $\varepsilon$-$\delta$ limit $\lim_{x \to c} f(x) = L$ of $f : A \to \mathbb{R}$ at a limit point $c$ of $A$: the $\varepsilon$-$\delta$ 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 $\varepsilon$-$\delta$ limit $\lim_{x \to c} f(x) = L$ of $f : A \to \mathbb{R}$ at a limit point $c$ of $A$, since $c \notin A^{-} \cup A^{+}$: the point $c$ belongs to neither $(-\infty, c)$ nor $(c, \infty)$.

• 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.