Goal: To understand rational functions where both
the numerator and denominator vanish.
The key to analyzing a rational function
at, or near, x = a when P(a) = 0 and Q(a) = 0
is the fol lowing theorem from algebra:
The Factor Theorem . Suppose P(x) is a polynomial and a is a root of P(x).
That is, P(a) = 0.
Then P(x) factors as P(x) = (x − a)Q(x), where Q(x) is another polynomial.
For example, let P(x) = x3 + 8. It’s easy to see that P(−2) = 0, since
P(−2) = (−2)3 + 8 = −8 + 8 = 0,
so by the above theorem, x−(−2) = x+2 is a factor of P(x). Using long division
we can EXPLICITLY
factor x + 2 out of P(x) to find that x3 + 8 = (x + 2)(x2 − 2x + 4).
1. Use the Factor Theorem to simplify the rational function
and discover that
its graph is a line missing a single point.
2. Use the Factor Theorem to see how to algebraically simplify each of the
following rational
functions. Sketch the graph for each function.
3. Use the graphs on the previous page to evaluate each of the following limits:
4. Explain how the Factor Theorem might be used to evaluate a limit
, where P(a) = 0
and Q(a) = 0, without first graphing the function
5. Use your observation in Problem 4 to evaluate the limit
.
