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Unformatted text preview: eorem 9.2. In the same way the
index k in the series
k=1 can never equal the upper limit ∞, but rather, ranges through all of the natural numbers, the index
k in the union
(2k + 1)π (2k + 3)π
k=−∞ can never actually be ∞ or −∞, but rather, this conveys the idea that k ranges through all of the
integers. Now that we have painstakingly determined the domain of F (t) = sec(t), it is time to
discuss the range. Once again, we appeal to the deﬁnition F (t) = sec(t) = cos(t) . The range of
f (t) = cos(t) is [−1, 1], and since F (t) = sec(t) is undeﬁned when cos(t) = 0, we split our discussion
into two cases: when 0 < cos(t) ≤ 1 and when −1 ≤ cos(t) < 0. If 0 < cos(t) ≤ 1, then we can
divide the inequality cos(t) ≤ 1 by cos(t) to obtain sec(t) = cos(t) ≥ 1. Moreover,using the notation
introduced in Section 4.2, we have that as cos(t) → 0+ , sec(t) = cos(t) ≈ very small (+) ≈ very big (+).
In other words, as cos(t) → 0+ , sec(t) → ∞. If, on the other hand, if −1 ≤ cos(t) < 0, then divid...
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