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Unformatted text preview: at all, both of these values of x still cause trouble in the denominator, and so, by
Theorem 4.1, x = −2 and x = 2 are vertical asymptotes of the graph. We can actually go
a step farther at this point and determine exactly how the graph approaches the asymptote
near each of these values. Though not absolutely necessary,4 it is good practice for those
heading oﬀ to Calculus. For the discussion that follows, it is best to use the factored form of
f (x) = (x−2)(x+2) .
• The behavior of y = f (x) as x → −2: Suppose x → −2− . If we were to build a table of
values, we’d use x-values a little less than −2, say −2.1, −2.01 and −2.001. While there
is no harm in actually building a table like we did in Section 4.1, we want to develop a
‘number sense’ here. Let’s think about each factor in the formula of f (x) as we imagine
substituting a number like x = −2.000001 into f (x). The quantity 3x would be very
close to −6, the quantity (x − 2) would be very close to −4, and the factor (x + 2) would
be very close to 0. More speciﬁcally, (x + 2) would be a little less than 0, in this case,
−0.000001. We will call such a number a ‘very small (−)’, ‘very small’ meaning close to
zero in absolute value. So, mentally, as x → −2− , we estimate
f (x) = 3x
(x − 2)(x + 2)
(−4) (very small (−))
2 (very small (−)) Now, the closer x gets to −2, the smaller (x + 2) will become, and so even thoug...
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