---
title: "Computing Limits with Algebraic Simplification"
slug: computing-limits-with-algebraic-simplification
type: evergreen
stime: 2025-01-15 @ 12:00AM
status: evergreen
certainty: certain
importance: 7
tags: [calculus, limits]
---

When direct substitution leads to an indeterminate form like $\frac{0}{0}$, simplify the expression algebraically.

**Example:**

$$
\lim_{x \to 3} \frac{x^2 - 9}{x - 3}
$$

Steps:

1. Factor the numerator:

$$
\frac{(x - 3)(x + 3)}{x - 3}
$$

2. Cancel $(x - 3)$:

$$
x + 3
$$

3. Plug in $x = 3$:

$$
3 + 3 = 6
$$

The simplification resolves the indeterminate form and reveals the limit.

## Related

- [Computing Basic Limits - Direct Substitution](computing-basic-limits-direct-substitution.md)
- [Limit with Difference of Cubes](limit-with-difference-of-cubes.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)