---
title: "Limit with Difference of Cubes"
slug: limit-with-difference-of-cubes
type: evergreen
stime: 2025-01-15 @ 12:00AM
status: evergreen
certainty: certain
importance: 5
tags: [calculus, limits]
---

When evaluating a limit like $\frac{x^3 - a^3}{x - a}$, use the **difference of cubes identity**:

$$
x^3 - a^3 = (x - a)(x^2 + ax + a^2)
$$

**Example:**

$$
\lim_{x \to 2} \frac{x^3 - 8}{x - 2}
$$

1. Recognize $8 = 2^3$
2. Apply identity:

$$
\frac{(x - 2)(x^2 + 2x + 4)}{x - 2}
$$

3. Cancel $x - 2$
4. Evaluate at $x = 2$: $4 + 4 + 4 = 12$

This method avoids undefined division and simplifies the expression cleanly.

## Related

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