Limits by Factoring Calculator
Enter numerator and denominator coefficients with a point of evaluation to resolve 0/0 indeterminate forms through algebraic factoring.
Limit Result
Step-by-Step Solution
What Are Limits by Factoring?
When you substitute x = a directly into a rational function and get 0/0, the function has a "hole" — a removable discontinuity. The limit still exists, but you need algebraic manipulation to find it. Factoring both the numerator and denominator often reveals a common factor of (x − a) that cancels out, leaving a simplified expression where direct substitution works.
For example, lim(x→3) of (x² − 5x + 6)/(x² − 4x + 3) gives 0/0 directly. Factoring yields (x−2)(x−3) / (x−1)(x−3). Cancel (x−3) to get (x−2)/(x−1). Now substituting x=3 gives 1/2. The limit is 1/2, even though the original function is undefined at x=3.
Indeterminate Forms Overview
Removable
Factor and cancel
Infinite
Limit is ±∞ or DNE
Direct
No factoring needed
How to Use the Limits by Factoring Calculator
Resolve 0/0 limits in three steps.
Enter Num/Den Coefficients
Type the quadratic coefficients for both numerator and denominator, plus the x-value where the limit is taken.
Click "Evaluate Limit"
The engine tests for 0/0, factors both polynomials, cancels common factors, and substitutes.
See the Limit Value
Get the final limit value, the before/after cancellation, and the complete step-by-step solution.
Worked Examples
Four limit problems resolved through factoring.
1lim(x→3) (x²−5x+6)/(x²−4x+3)+−
2lim(x→2) (x²−4)/(x²−2x)+−
3lim(x→−1) (x²−1)/(x+1)+−
4lim(x→1) (x²+x−2)/(x²−1)+−
Limit Laws Reference
Key limit laws and techniques for evaluating limits.
| Law | Formula | When to Use |
|---|---|---|
| Direct Substitution | lim f(x) = f(a) | f is continuous at a |
| Factoring | Cancel (x−a) from 0/0 | Polynomial 0/0 forms |
| L'Hôpital's Rule | lim f/g = lim f'/g' | When factoring fails |
| Rationalization | Multiply by conjugate | Square root expressions |
Frequently Asked Questions
Common questions about evaluating limits by factoring.
When do you use factoring to evaluate a limit?+−
Use factoring when direct substitution gives 0/0, an indeterminate form. Factor both numerator and denominator, cancel the common factor, then substitute again.
What is an indeterminate form?+−
An indeterminate form like 0/0 means direct substitution doesn't reveal the limit. The expression needs algebraic manipulation — factoring, rationalizing, or L'Hôpital's rule — before the limit can be evaluated.
Can every 0/0 limit be solved by factoring?+−
Not all, but many can. Factoring works when the numerator and denominator share a common factor (x − a). For expressions involving radicals or transcendental functions, other techniques may be needed.
What if the limit still gives 0/0 after canceling?+−
If canceling one common factor still yields 0/0, the expressions may share multiple factors of (x − a). Factor again or apply L'Hôpital's rule.
How does this relate to continuity?+−
If lim f(x) as x→a equals f(a), the function is continuous at a. A removable discontinuity occurs when the limit exists but f(a) is undefined — exactly the situation factoring resolves.
Can this calculator handle rational expressions only?+−
Yes — this calculator focuses on polynomial rational expressions (polynomial/polynomial) where factoring is the primary technique for resolving 0/0 forms.
Related Tools Showcase
Explore 4 complementary calculators tailored to expand your mathematical analysis.