---
tag: 0909PWZR62
kind: definition
name: One-sided limit
aliases: [left-limit, right-limit]
msc: [26A03]
defines: 00BE2J8H05
uses: [0TXTSEMHR5]
sources:
  - url: https://en.wikipedia.org/wiki/One-sided_limit
    locator: Formal definition
---

## Definition

Let $D \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$, and let $a$ be a limit point of $D \cap (a, \infty)$.
A real number $L$ is the right limit of $f$ at $a$ if the following holds.
For each $\varepsilon > 0$ there is $\delta > 0$ with $|f(x) - L| < \varepsilon$ for all $x \in D$ with $a < x < a + \delta$.
The notation is $\lim_{x \to a^+} f(x) = L$.
The left limit $\lim_{x \to a^-} f(x)$ is defined in the same way, with $a - \delta < x < a$.

## Remarks

The right limit is the [[limit|0TXTSEMHR5]] of the restriction of $f$ to $D \cap (a, \infty)$.
If $a$ is a limit point of points on both sides, then $\lim_{x \to a} f(x) = L$ if and only if both one-sided limits exist and equal $L$.
For the sign function at $0$, the right limit is $1$ and the left limit is $-1$.

## Explanations

### L1 human

A one-sided limit only looks at points on one side of $a$.
A function with a jump at $a$ has two different one-sided limits there.