---
title: "When a Limit Exists"
slug: when-a-limit-exists
type: evergreen
stime: 2025-01-15 @ 12:00AM
status: evergreen
certainty: certain
importance: 6
tags: [calculus, limits]
---

A limit $\lim_{x \to a} f(x) = L$ exists only if the function approaches the same value from both sides:

$$
\lim_{x \to a^-} f(x) = L \quad \text{and} \quad \lim_{x \to a^+} f(x) = L
$$

- The **left-hand limit** ($x \to a^-$) looks at values from the left
- The **right-hand limit** ($x \to a^+$) looks at values from the right

If these two limits agree, the two-sided limit exists. If they differ, the two-sided limit **does not exist** (DNE).

This is common at jump discontinuities or in absolute value functions where behavior changes depending on direction.

## Related To This Note

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