Limits by Factoring: Step-by-Step Guide
When direct substitution gives the indeterminate form 0/0, factoring often resolves the limit. The key idea is that the numerator and denominator share a common factor (x - a) that can be canceled, leaving a function whose limit can be found by direct substitution.
Math Racer skill guide · Calculus 1 · AP Calc AB / College
How to solve it, step by step
- Try direct substitution to confirm 0/0. Plug x = a into the expression. If you get 0/0, factoring is a likely method. (If you get a defined value, you are done.)\[\lim_{x \to a} \frac{f(x)}{g(x)} \to \frac{0}{0}\]
- Factor numerator and denominator. Factor each polynomial fully. Look for the common factor (x - a). Useful patterns: difference of squares a^2 - b^2 = (a-b)(a+b), trinomials, and grouping.\[\frac{f(x)}{g(x)} = \frac{(x-a)\,p(x)}{(x-a)\,q(x)}\]
- Cancel the common factor. Cancel (x - a) from numerator and denominator. The new expression is equal to the original everywhere except at x = a, but limits ignore the value at x = a.\[\lim_{x \to a} \frac{p(x)}{q(x)}\]
- Substitute again. Plug x = a into the simplified expression. If it gives a finite value, that is the limit.\[\lim_{x \to a} \frac{p(x)}{q(x)} = \frac{p(a)}{q(a)}\]
Worked example
Evaluate lim x->2 (x^2 - 4)/(x - 2).
Step 1: Try direct substitution: (4 - 4)/(2 - 2) = 0/0, indeterminate, so factor.
Step 2: Factor the numerator using the difference of squares: x^2 - 4 = (x - 2)(x + 2).
Step 3: Rewrite the limit: lim x->2 ((x - 2)(x + 2))/(x - 2).
Step 4: Cancel (x - 2): lim x->2 (x + 2).
Step 5: Substitute x = 2: 2 + 2 = 4.
Step 6: Therefore the limit is 4. Verification: f(2.01) = (4.0401 - 4)/0.01 = 4.01, confirming the limit.
Tips and common mistakes
- Always confirm 0/0 first; if substitution gives a finite number, no factoring is needed.
- If x = a is a root of both numerator and denominator, (x - a) must be a common factor.
- Memorize key factoring patterns: a^2 - b^2 = (a-b)(a+b), a^3 - b^3 = (a-b)(a^2+ab+b^2), a^3 + b^3 = (a+b)(a^2-ab+b^2).
- For trinomials ax^2 + bx + c, factor by finding two numbers that multiply to ac and add to b.
- Polynomial long division or synthetic division can extract (x - a) from any polynomial that has a as a root.
- After canceling, you may still get 0/0 — repeat the process or try a different method like L'Hopital's.
- Plug in a value close to a (like 1.99) to sanity-check your final answer.
- Factoring works in conjunction with rationalization for limits involving square roots.