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Unformatted text preview: as we move to the far left.
• The behavior of y = f (x) as x → ∞: On the ﬂip side, we can imagine substituting very
large positive numbers in for x and looking at the behavior of f (x). For example, let
x = 1 billion. Proceeding as before, we get
f (1 billion) ≈ 3
≈ very small (+)
billion The larger the number we put in, the smaller the positive number we would get out. In
other words, as x → ∞, f (x) → 0+ , so the graph of y = f (x) is a little bit above the
x-axis as we look toward the far right.
Graphically, we have7
−1 6. Lastly, we construct a sign diagram for f (x). The x-values excluded from the domain of f
are x = ±2, and the only zero of f is x = 0. Displaying these appropriately on the number
line gives us four test intervals, and we choose the test values8 we x = −3, x = −1, x = 1,
and x = 3. We ﬁnd f (−3) is (−), f (−1) is (+), f (1) is (−), and f (3) is (+). Combining this
with our previous work, we get the graph of y = f (x) below.
7 As with the vertical asymptotes in...
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