One-Sided Limit¶

Definition¶

Let $f$ be a real-valued function and let $x_0 \in \mathbb{R}\text{.}$

Assume that there exists an $r > 0$ such that $f$ is defined for every $x \in (x_0, x_0 + r)\text{.}$ We say that $f$ has a right-hand limit $L \in \mathbb{R}$ at $x = x_0\text{,}$ and write

f(x_0^+) = \lim_{x \to x_0^+} f(x) = \lim_{x \downarrow x_0} f(x) = L,

if for every $\varepsilon > 0$ there exists a number $\delta > 0$ with $0 < \delta \le r$ such that

x_0 < x < x_0 + \delta \implies |f(x) - L| < \varepsilon.

Assume that there exists an $r > 0$ such that $f$ is defined for every $x \in (x_0 - r, x_0)\text{.}$ We say that $f$ has a left-hand limit $L \in \mathbb{R}$ at $x = x_0\text{,}$ and write

f(x_0^-) = \lim_{x \to x_0^-} f(x) = \lim_{x \uparrow x_0} f(x) = L,

if for every $\varepsilon > 0$ there exists a number $\delta > 0$ with $0 < \delta \le r$ such that

x_0 - \delta < x < x_0 \implies |f(x) - L| < \varepsilon.

Interpretation¶

One-sided limits describe the behaviour of a function as the input approaches a point from one direction only: for the right-hand limit only the values at $x>x_0$ count, for the left-hand limit only those at $x<x_0$ $x_{0}$ . As for the limit at a point, the value $f(x_0)$ plays no role.

Only one side of $x_0$ counts. For the jump, $L=2$ works from the right and $L=1$ from the left; switch sides and see each candidate fail on the other side.

Compared with the two-sided definition, the condition $0<|x-x_0|<\delta$ is cut in half to $x_0<x<x_0+\delta$ or $x_0-\delta<x<x_0$ $x_{0}$ , so a one-sided limit asks for less. If $f$ is defined on both sides of $x_0$ , the two-sided limit exists if and only if both one-sided limits exist and are equal, and then all three coincide. This is the standard way to show that a limit does not exist: find two one-sided limits that differ, as for the signum function at $0$ .

One-sided limits make limits possible at an endpoint of an interval, where only one side belongs to the domain, and they classify singularities: a jump has two finite but different one-sided limits, and a vertical asymptote needs only one infinite one-sided limit. They also define one-sided continuity.

The notation $f(x_0^+)$ stands for a limit, not for a value of $f$ . In general $f(x_0^-)$ , $f(x_0)$ and $f(x_0^+)$ can be three different numbers: for the signum function at $0$ they are $-1$ , $0$ and $1$ .

One-sided limits return later in one-sided derivatives.

Key Results¶

One-sided limits may exist independently of each other. Infinite one-sided limits are defined analogously: $\displaystyle\lim_{x\to x_0^+}f(x)=+\infty$ means that for every $M\in\mathbb{R}$ there exists $\delta>0$ such that $x_0<x<x_0+\delta\implies f(x)>M\text{;}$ the cases with $-\infty$ or $x\to x_0^-$ are analogous (see divergence to infinity).
Equality of both one-sided limits implies existence of the two-sided limit.
One-sided limits are essential for defining one-sided continuity and derivatives.

Examples¶

The signum function has distinct one-sided limits at the origin.

The signum function with its two horizontal branches at -1 and 1 and the value 0 at the origin; points approaching 0 from the right give values 1, and from the left give values -1.

Along $x_n\to0^+$ the values are $1$ , along $y_n\to0^-$ they are $-1$ : $\lim_{x\to0^+}\operatorname{sgn}(x)=1\neq-1=\lim_{x\to0^-}\operatorname{sgn}(x)$ $lim_{x \to 0^{-}} sgn (x)$ .

Prerequisites¶

Limit at a Point

Sources¶

Math I: Calculus · Lecture 5

Aliases: right-hand limit, left-hand limit, right limit, left limit Symbols: delta, element of, epsilon, infinity, real numbers

Last updated: 20 October 2026 03:02