Factoring Techniques for Limits
Factoring is your best friend when dealing with 0/0 limits. Let's master the factoring techniques you'll use most often in calculus.
Common Factoring Patterns
Difference of Squares
Example: x^2 - 9 = (x + 3)(x - 3)
a^2 - b^2 = (a + b)(a - b)Example: x^2 - 9 = (x + 3)(x - 3)
Sum/Difference of Cubes
Example: x^3 - 8 = (x - 2)(x^2 + 2x + 4)
a^3 +/- b^3 = (a +/- b)(a^2 -/+ ab + b^2)Example: x^3 - 8 = (x - 2)(x^2 + 2x + 4)
Quadratic
Example: x^2 + 5x + 6 = (x + 2)(x + 3)
x^2 + bx + c = (x + m)(x + n) where mn = c, m + n = bExample: x^2 + 5x + 6 = (x + 2)(x + 3)
Difference of Cubes
Medium
Find lim[x->2] (x^3 - 8)/(x - 2)
1
Check direct substitution
(8 - 8)/(2 - 2) = 0/0 - Indeterminate!
2
Recognize the pattern
x^3 - 8 = x^3 - 2^3 is a difference of cubes
3
Apply the formula
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
x^3 - 8 = (x - 2)(x^2 + 2x + 4)
4
Cancel and substitute
(x - 2)(x^2 + 2x + 4)/(x - 2) = x^2 + 2x + 4
lim[x->2] (x^2 + 2x + 4) = 4 + 4 + 4 = 12
Answer:
12
Practice Problems
1. Find lim[x->4] (x^2 - 16)/(x - 4)
x^2 - 16 = (x+4)(x-4). Cancel (x-4) to get x+4. At x=4: 4+4=8
2. Find lim[x->3] (x^3 - 27)/(x - 3)
x^3 - 27 = (x-3)(x^2+3x+9). Cancel to get x^2+3x+9 = 9+9+9=27
Key Takeaways
Difference of squares: a^2 - b^2 = (a+b)(a-b)
Difference of cubes: a^3 - b^3 = (a-b)(a^2+ab+b^2)
For quadratics, find two numbers that multiply to c and add to b