Evaluating a Limit by Factoring

Every board from the lesson, so you can read it at your own pace or copy it into your notes.

1. 1. Try substitution0:00
   1. $$\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$$
   2. $$\frac{3^2 - 9}{3 - 3} = \frac{0}{0}$$
      indeterminate: more work needed
2. 2. Factor and cancel0:14
   1. $$\frac{x^2 - 9}{x - 3} = \frac{(x - 3)(x + 3)}{x - 3}$$
   2. $$= x + 3 \quad (x \ne 3)$$
   3. $$\lim_{x \to 3} (x + 3) = 6$$
3. 3. The hole in the graph0:33

4. 4. Recap0:45
   • ✓$\frac{0}{0}$ means: simplify first.
   • ✓Factor, cancel the common factor, substitute.
   • ✓The limit can exist where the function does not.

Evaluate limits algebraically and decide continuity.

• Limit Calculator: try the same kind of problem with your own numbers.
• More calculus practice problems
• Limits Study Guide (study guide)
• AP Calculus AB Unit 1: Limits and Continuity: the AP unit this lesson supports.
• Calculus calculators and topics
• Step-by-step solver: check your own answer with hints.

Unofficial study material. This site is an independent educational resource and is not affiliated with or endorsed by the College Board. AP® is a trademark registered by the College Board, which is not involved in producing this content.

▶0:50

Calculuseasy4 chapters

The Power Rule for Derivatives

Differentiate 3x⁴ − 2x² + 7x term by term with the power rule, then use the derivative to find the slope of the tangent line at x = 1.

▶0:47

Calculusmedium4 chapters

The Definite Integral as Area

Find the area under y = x² + 1 from 0 to 2 with an antiderivative and the Fundamental Theorem of Calculus: 14/3, about 4.67 square units.

▶0:49

Calculusmedium4 chapters

The Chain Rule

Differentiate (3x² + 1)⁴ with the chain rule: derivative of the outside, times derivative of the inside, giving 24x(3x² + 1)³.

See every lesson in the gallery