---
title: "Limit with Absolute Value - Jump Discontinuity"
slug: limit-with-absolute-value-jump-discontinuity
type: evergreen
stime: 2025-01-15 @ 12:00AM
status: growing
certainty: certain
importance: 5
tags: [calculus, limits]
---

**Example:**

$$
\lim_{x \to 3} \frac{|x - 3|}{x - 3}
$$

- From the left ($x < 3$), $|x - 3| = -(x - 3)$ $\to$ value = $-1$
- From the right ($x > 3$), $|x - 3| = x - 3$ $\to$ value = $1$

Since the two one-sided limits aren't equal, the limit **does not exist**.

This is a classic example of a **jump discontinuity**, where the function abruptly switches values at a point.

## Related

- [When a Limit Exists](when-a-limit-exists.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)
- [Definition of Absolute Value](definition-of-absolute-value.md)
- [Notation of a Limit](notation-of-a-limit.md)
- [Differential Calculus](differential-calculus.md)
- [Computing Basic Limits - Direct Substitution](computing-basic-limits-direct-substitution.md)
- [Function Notation](function-notation.md)
- [Computing Limits with Algebraic Simplification](computing-limits-with-algebraic-simplification.md)
- [Limit with Difference of Cubes](limit-with-difference-of-cubes.md)