Unformatted text preview: havior of y = f (x) as x → −∞: If we were to make a table of values to discuss
the behavior of f as x → −∞, we would substitute very ‘large’ negative numbers in for
x, say, for example, x = −1 billion. The numerator 3x would then be −3 billion, whereas
5 The actual retail value of f (−2.000001) is approximately −1,500,000.
We have deliberately left oﬀ the labels on the y -axis because we know only the behavior near x = ±2, not the
actual function values.
6 250 Rational Functions
the denominator x2 − 4 would be (−1 billion)2 − 4, which is pretty much the same as
1(billion)2 . Hence,
f (−1 billion) ≈ −3 billion
≈ very small (−)
billion Notice that if we substituted in x = −1 trillion, essentially the same kind of cancellation
would happen, and we would be left with an even ‘smaller’ negative number. This not
only conﬁrms the fact that as x → −∞, f (x) → 0, it tells us that f (x) → 0− . In other
words, the graph of y = f (x) is a little bit below the x-axis...
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