College Algebra, 2013a

326 Rational Functions
so as x →−2 − , g(x) →∞. On the flip side, as x →−2 + ,weget
g(x) ≈
9
≈ very big (−)
very small (−)
so g(x) →−∞.
ˆ 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) ≈
(1)(4)
(verysmall(−))(5) ≈ 4
≈ very big (−)
very small (−)
Hence, as x → 3 − , g(x) →−∞.Asx → 3 + ,weget
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 the graph of g by taking the
ratio of the leading terms coefficients, y = 2 1
= 2. However, if we take the time to do a
more detailed analysis, we will be able to reveal some ‘hidden’ behavior which would be lost
(
otherwise. 11 As in the discussion following Theorem 4.2, we use the result of the long division
2x 2 − 3x − 5 ) ÷ ( x 2 − x − 6 ) to rewrite g(x) = 2x2 −3x−5
as g(x) =2− x−7 . We focus our
x 2 −x−6 x 2 −x−6
attention on the term
x−7
x 2 −x−6 .
11 That is, if you use a calculator to graph. Once again, Calculus is the ultimate graphing power tool.