Chapter 7 Rational Functions - College of the Redwoods

606 Chapter 7 Rational Functions
and x = 1. We evaluate y = 1/x at x = 1/2, x = 1/4, and x = 1/8, as shown in the
table in Figure 3, then plot the resulting points.
y
10
x
10
x y = 1/x
1/2 2
1/4 4
1/8 8
Figure 3. At the right is a table of points satisfying the equation y =
1/x. These points are plotted as solid dots on the graph at the left.
Note that the x-values in the table in Figure 3 approach zero from the right, then
note that the corresponding y-values are getting larger and larger. We could continue
in this vein, adding points. For example, if x = 1/16, then y = 16. If x = 1/32, then
y = 32. If x = 1/64, then y = 64. Each time we halve our value of x, the resulting
value of x is closer to zero, and the corresponding y-value doubles in size. Calculus
students describe this behavior with the notation
lim y = lim 1
= ∞. (8)
x→0 + x→0 + x
That is, as “x approaches zero from the right, the value of y grows to infinity.” This is
evident in the graph in Figure 3, where we see the plotted points move closer to the
vertical axis while at the same time moving upward without bound.
A similar thing happens on the other side of the vertical axis, as shown in Figure 4.
y
10
x
10
x y = 1/x
−1/2 −2
−1/4 −4
−1/8 −8
Figure 4. At the right is a table of points satisfying the equation y =
1/x. These points are plotted as solid dots on the graph at the left.
Version: Fall 2007