This preview shows page 1. Sign up to view the full content.
Unformatted text preview: x2x−7 6 as x → ∞, we imagine
substituting x = 1 billion and, going through the usual mental routine, ﬁnd
≈ very small (+)
x2 − x − 6
Hence, g (x) ≈ 2 − very small (+), in other words, the graph of y = g (x) is just below
the line y = 2 as x → ∞.
On y = g (x), we have (again, without labels on the x-axis)
y 1 x
−1 6. Finally we construct our sign diagram. We place an ‘ ’ above x = −2 and x = 3, and a ‘0’
above x = 5 and x = −1. Choosing test values in the test intervals gives us f (x) is (+) on
the intervals (−∞, −2), −1, 2 , and (3, ∞), and (−) on the intervals (−2, −1) and 2 , 3 .
As we piece together all of the information, we note that the graph must cross the horizontal
asymptote at some point after x = 3 in order for it to approach y = 2 from underneath. This
is the subtlety that we would have missed had we skipped the long division and subsequent
end behavior analysis. We can, in fact, ﬁnd exactly when the graph crosses y = 2. As a result
of the long divisi...
View Full Document