---
title: "Notation of a Limit"
slug: notation-of-a-limit
type: evergreen
stime: 2025-01-15 @ 12:00AM
status: evergreen
certainty: certain
importance: 7
tags: [calculus, limits]
---

A limit describes what a function approaches as the input gets close to a certain value. It's written as:

$$
\lim_{x \to a} f(x) = L
$$

This reads: "The limit of $f(x)$ as $x$ approaches $a$ is $L$."

Formally, it means: for every $\epsilon > 0$, there exists a $\delta > 0$ such that if $0 < |x - a| < \delta$, then $|f(x) - L| < \epsilon$.

This $\epsilon$-$\delta$ definition is the rigorous foundation of limits, though intuitive understanding comes first.

Limits let us talk about function behavior near a point, even if $f(a)$ is undefined. They are critical to defining continuity and derivatives.

## Related

- [When a Limit Exists](when-a-limit-exists.md)
- [Computing Basic Limits - Direct Substitution](computing-basic-limits-direct-substitution.md)
- [Computing Limits with Algebraic Simplification](computing-limits-with-algebraic-simplification.md)
- [Function Notation](function-notation.md)
- [Limit with Absolute Value - Jump Discontinuity](limit-with-absolute-value-jump-discontinuity.md)
- [Limit with Absolute Value - Piecewise Behavior](limit-with-absolute-value-piecewise-behavior.md)
- [Limit with Difference of Cubes](limit-with-difference-of-cubes.md)
- [Differential Calculus](differential-calculus.md)
- [Gaussian Integral](gaussian-integral.md)
- [Definition of Absolute Value](definition-of-absolute-value.md)
- [Absolute Value as Piecewise Function](absolute-value-as-piecewise-function.md)