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Unformatted text preview: (7.5, ∞). We ﬁnd r(x) < 0 on (7.5, ∞).
1 There is no asymptote at x = 1 since the graph is well behaved near x = 1. According to Theorem 4.1, there
must be a hole there. 270 Rational Functions
(+) 0 (−)
0 7.5 In the context of the problem, x represents the number of PortaBoy games systems produced
and AC (x) is the average cost to produce each system. Solving AC (x) < 100 means we are
trying to ﬁnd how many systems we need to produce so that the average cost is less than $100
per system. Our solution, (7.5, ∞) tells us that we need to produce more than 7.5 systems to
achieve this. Since it doesn’t make sense to produce half a system, our ﬁnal answer is [8, ∞).
4. We can apply Theorem 4.2 to AC (x) and we ﬁnd y = 80 is a horizontal asymptote to the
graph of y = AC (x). To more precisely determine the behavior of AC (x) as x → ∞, we
ﬁrst use long division2 and rewrite AC (x) = 80 + 150 . As x → ∞, 150 → 0+ , which means
AC (x) ≈ 80 + very small (+). Thus the average cost per system is getting closer to $80
per system. If we set AC (x)...
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