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**Unformatted text preview: **f Vertical Asymptotes and Holes:a Suppose r is a rational function
( x)
which can be written as r(x) = p(x) where p and q have no common zeros.b Let c be a real number
q
which is not in the domain of r.
(c)
• If q (c) = 0, then the graph of y = r(x) has a hole at c, p(c) .
q • If q (c) = 0, then the the line x = c is a vertical asymptote of the graph of y = r(x).
a
b Or, ‘How to tell your asymptote from a hole in the graph.’
In other words, r(x) is in lowest terms. In English, Theorem 4.1 says if x = c is not in the domain of r but, when we simplify r(x), it
no longer makes the denominator 0, then we have a hole at x = c. Otherwise, we have a vertical
asymptote.
Example 4.1.2. Find the vertical asymptotes of, and/or holes in, the graphs of the following
rational functions. Verify your answers using a graphing calculator.
1. f (x) = 2x
2−3
x 3. h(x) = x2 − x − 6
x2 + 9 2. g (x) = x2 − x − 6
x2 − 9 4. r(x) = x2 − x − 6
x2 + 4 x + 4 Solution.
1. To use Theorem 4.1, we ﬁrst ﬁnd all of the √ numbers which aren’t in the domain of f . To
real
2 − 3 = 0 and get x = ± 3. Since the expression f (x) is in lowest terms,
do so, we solve x
√
√
there is no...

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