Unformatted text preview: ce n − m is a positive
number and r(x) ≈ ax b , which becomes unbounded as x → ±∞. As we said before, if a rational
function has a horizontal asymptote, then it will have only one. (This is not true for other types
of functions we shall see in later chapters.) Example 4.1.4. Determine the horizontal asymptotes, if any, of the graphs of the following functions. Verify your answers using a graphing calculator.
1. f (x) = 5x
+1 x2 2. g (x) = x2 − 4
x+1 3. h(x) = 6x3 − 3x + 1
5 − 2x3 Solution.
1. The numerator of f (x) is 5x, which is degree 1. The denominator of f (x) is x2 + 1, which
is degree 2. Applying Theorem 4.2, y = 0 is the horizontal asymptote. Sure enough, as
x → ±∞, the graph of y = f (x) gets closer and closer to the x-axis.
2. The numerator of g (x), x2 − 4, is degree 2, but the degree of the denominator, x + 1, is degree
1. By Theorem 4.2, there is no horizontal asymptote. From the graph, we see the graph of
Note that as x → −∞, f (x) → 2+ , whereas as x → ∞, f (x) → 2− . We write f (x) → 2 if we are unconcerned
from which direction...
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