Master the factoring and canceling technique to solve limits of rational functions that result in the indeterminate form 0/0. This method involves factoring the numerator and denominator to cancel out the term causing the zero.
When is the factoring and canceling technique most useful for evaluating limits?
It is most effective for limits of rational functions (polynomials divided by polynomials) that yield the indeterminate form \(\frac{0}{0}\) upon direct substitution.
What is the primary goal of the factoring and canceling technique?
To algebraically eliminate the 'hole' in the graph. This is done by factoring both the numerator and denominator to find a common factor, \((x-c)\), which can be canceled out.
$$\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$$
Factoring the numerator gives \(\lim_{x \to 2} \frac{(x-2)(x+2)}{x-2}\). After canceling the \((x-2)\) terms, we evaluate \(\lim_{x \to 2} (x+2) = 2+2 = 4\).
Why are we mathematically allowed to cancel the term \((x-c)\) when evaluating \(\lim_{x \to c}\)?
The definition of a limit examines the behavior of a function as \(x\) gets arbitrarily close to \(c\), but not when \(x\) is equal to \(c\). Therefore, \(x-c \neq 0\), and division by this term is valid.
What is the factorization for a difference of squares, \(a^2 - b^2\)?
$$a^2 - b^2 = (a-b)(a+b)$$
What is the factorization for a difference of cubes, \(a^3 - b^3\)?
$$a^3 - b^3 = (a-b)(a^2 + ab + b^2)$$
What is the factorization for a sum of cubes, \(a^3 + b^3\)?
What should you do if a polynomial in a limit problem is difficult to factor by inspection?
If the limit is of the form \(\lim_{x \to c}\) and results in \(\frac{0}{0}\), you know that \((x-c)\) must be a factor of the polynomial. You can use polynomial long division or synthetic division to find the other factors.
