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**Unformatted text preview: ** can be ‘plugged’ when embarking on a more advanced
study of continuity.
5 236 Rational Functions Neither x = −1 nor x = 1 are in the domain of h, yet we see the behavior of the graph of y = h(x)
is drastically diﬀerent near these points. The reason for this lies in the second to last step when we
x−1)(x−1)
simpliﬁed the formula for h(x) in Example 4.1.1. We had h(x) = (2x+1)(x−1) . The reason x = −1 is
(
not in the domain of h is because the factor (x + 1) appears in the denominator of h(x); similarly,
x = 1 is not in the domain of h because of the factor (x − 1) in the denominator of h(x). The major
diﬀerence between these two factors is that (x − 1) cancels with a factor in the numerator whereas
(x + 1) does not. Loosely speaking, the trouble caused by (x − 1) in the denominator is canceled
away while the factor (x + 1) remains to cause mischief. This is why the graph of y = h(x) has
a vertical asymptote at x = −1 but only a hole at x = 1. These observations are generalized and
summarized in the theorem below, whose proof is found in Calculus.
Theorem 4.1. Location o...

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