Stitz-Zeager_College_Algebra_e-book

# Recall that the intervals where hx 0 or correspond

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Unformatted text preview: ound, x → −∞, and as x becomes large without bound, x → ∞. Making tables of values, we ﬁnd x −10 −100 −1000 −10000 f (x) (x, f (x)) ≈ 2.3333 ≈ (−10, 2.3333) ≈ 2.0303 ≈ (−100, 2.0303) ≈ 2.0030 ≈ (−1000, 2.0030) ≈ 2.0003 ≈ (−10000, 2.0003) x 10 100 1000 10000 f (x) (x, f (x)) ≈ 1.7273 ≈ (10, 1.7273) ≈ 1.9703 ≈ (100, 1.9703) ≈ 1.9970 ≈ (1000, 1.9970) ≈ 1.9997 ≈ (10000, 1.9997) From the tables, we see as x → −∞, f (x) → 2+ and as x → ∞, f (x) → 2− . Here the ‘+’ means ‘from above’ and the ‘−’ means ‘from below’. In this case, we say the graph of y = f (x) has a horizontal asymptote of y = 2. This means that the end behavior of f resembles the horizontal line y = 2, which explains the ‘leveling oﬀ’ behavior we see in the calculator’s graph. We formalize the concepts of vertical and horizontal asymptotes in the following deﬁnitions. Definition 4.2. The line x = c is called a vertical asymp...
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