Unformatted text preview: Keeping in mind g (x) = (2x−3)(x+2) , we proceed to our
(
analysis near each of these values.
• The behavior of y = g (x) as x → −2: As x → −2− , we imagine substituting a number
a little bit less than −2. We have
g (x) ≈
9 (−9)(−1)
9
≈
≈ very big (+)
(−5)(very small (−))
very small (+) And Jeﬀ doesn’t think much of it to begin with... 252 Rational Functions
so as x → −2− , g (x) → ∞. On the ﬂip side, as x → −2+ , we get
g (x) 9
≈ very big (−)
very small (−) so g (x) → −∞.
• The behavior of y = g (x) as x → 3: As x → 3− , we imagine plugging in a number just
shy of 3. We have
g (x) ≈ 4
(1)(4)
≈
≈ very big (−)
( very small (−))(5)
very small (−) Hence, as x → 3− , g (x) → −∞. As x → 3+ , we get
g (x) ≈ 4
≈ very big (+)
very small (+) so g (x) → ∞.
Graphically, we have (again, without labels on the y axis)
y −3 −1 1 2 4 x 5. Since the degrees of the numerator and denominator of g (x) are the same, we know from
Theorem 4.2 that we can ﬁnd the horizontal asymptote of...
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 Fall '13
 Wong
 Algebra, Trigonometry, Cartesian Coordinate System, The Land, The Waves, René Descartes, Euclidean geometry

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