James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

134 Chapter 2 Limits and Derivatives
The next example shows that by using infinite limits at infinity, together with intercepts,
we can get a rough idea of the graph of a polynomial without having to plot a large
number of points.
y
_1 0
1 2 x
_16
FIGURE 13
y − sx 2 2d 4 sx 1 1d 3 sx 2 1d
ExamplE 12 Sketch the graph of y − sx 2 2d 4 sx 1 1d 3 sx 2 1d by finding its intercepts
and its limits as x l ` and as x l 2`.
SOLUTION The y-intercept is f s0d − s22d 4 s1d 3 s21d − 216 and the x-intercepts are
found by setting y − 0: x − 2, 21, 1. Notice that since sx 2 2d 4 is never negative,
the function doesn’t change sign at 2; thus the graph doesn’t cross the x-axis at 2. The
graph crosses the axis at 21 and 1.
When x is large positive, all three factors are large, so
lim sx 2
x l ` 2d4 sx 1 1d 3 sx 2 1d − `
When x is large negative, the first factor is large positive and the second and third factors
are both large negative, so
lim sx 2
x l2` 2d4 sx 1 1d 3 sx 2 1d − `
Combining this information, we give a rough sketch of the graph in Figure 13.
n
Precise Definitions
Definition 1 can be stated precisely as follows.
7 Precise Definition of a Limit at Infinity Let f be a function defined on some
interval sa, `d. Then
lim
x l ` f sxd − L
means that for every « . 0 there is a corresponding number N such that
if x . N then | f sxd 2 L | , «
In words, this says that the values of f sxd can be made arbitrarily close to L (within a
distance «, where « is any positive number) by requiring x to be sufficiently large (larger
than N, where N depends on «). Graphically it says that by keeping x large enough
(larger than some number N) we can make the graph of f lie between the given horizontal
lines y − L 2 « and y − L 1 « as in Figure 14. This must be true no matter how
small we choose «.
y
L
y=L+∑
∑
∑
y=L-∑
y=ƒ
ƒ is
in here
FIGURE 14
lim f sxd − L
x l `
0
N
when x is in here
x
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