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

90 Chapter 2 Limits and Derivatives
When we say a number is “large negative,”
we mean that it is negative but its
magnitude (absolute value) is large.
y
x=a
0 a
x
y=ƒ
A similar sort of limit, for functions that become large negative as x gets close to a, is
defined in Definition 5 and is illustrated in Figure 13.
5 Definition Let f be a function defined on both sides of a, except possibly at
a itself. Then
lim
x l a
f sxd − 2`
means that the values of f sxd can be made arbitrarily large negative by taking x
sufficiently close to a, but not equal to a.
figure 13
lim f sxd − 2`
x l a
The symbol lim x l a f sxd − 2` can be read as “the limit of f sxd, as x approaches a, is
negative infinity” or “ f sxd decreases without bound as x approaches a.” As an example
we have
2D lim
x l0S2 1 − 2`
x
Similar definitions can be given for the one-sided infinite limits
lim f sxd − `
x la 2
lim f sxd − 2` lim
x la2 lim f sxd − `
x la 1
f sxd − 2`
x la1 remembering that x l a 2 means that we consider only values of x that are less than a,
and similarly x l a 1 means that we consider only x . a. Illustrations of these four
cases are given in Figure 14.
y
y
y
y
0 a x
0
a
x
0 a x
0 a
x
(b) lim ƒ=`
(a) lim ƒ=`
x a _ x a +
FIGURE 14
(c) lim ƒ=_`
x
a _
(d) lim ƒ=_`
x a +
6 Definition The vertical line x − a is called a vertical asymptote of the
curve y − f sxd if at least one of the following statements is true:
lim f sxd − `
x la
lim f sxd − 2` lim
x la
lim f sxd − `
x la 2
f sxd − 2` lim
x la2 lim f sxd − `
x la 1
f sxd − 2`
x la1 For instance, the y-axis is a vertical asymptote of the curve y − 1yx 2 because
lim x l 0 s1yx 2 d − `. In Figure 14 the line x − a is a vertical asymptote in each of
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.