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Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4
Sullivan
128 Chapter 1 • Limits and Continuity
Teaching Tip
1
Note that when finding the limit of
x
2
as x approaches zero, the authors take
a tabular and graphical approach. There
is no algebraic manipulation that can be
1
performed to simplify the function , so a
x
2
table or graph is the preferred method for
this problem. Some students may prefer to
create a table mentally when a written table
is not required for the answer. For example,
since zero cannot be substituted in this
situation, the students may find it helpful
to think, What does 1 divided by a really
small positive number squared equal? Then
they should think, What does 1 divided by
a really small negative number squared
equal?
In both cases, the answer comes out to
be an exceptionally large number, which
grows increasingly large as x gets closer
and closer to zero.
Therefore. lim 1 =∞.
x→0
2
x
Teaching Tip
It may be helpful for the students to keep
these two limits in mind. They appear often
as the students find one-sided limits and
define vertical asymptotes.
1
lim
=∞
x→ 0
+ very small positive value
1
lim
=−∞
x→0
− very small negative value
Numerical examples help here as well:
1/0.1 = 10, 1/0.01 = 100, and so forth.
For example,
1
=
1 1
1
=
0.1 10
1
=
0.01 100
1
=
0.001 1000
1
=
0.0001 10,000
y
4
1 Investigate Infinite Limits
Consider the function f (x) = 1 . Table 11 lists values of f for selected numbers x that
x
2
are close to 0 and Figure 48 shows the graph of f .
Figure 48 f (x) = 1 x 2 As x approaches 0, the value of 1 x increases without bound. Since the value of 1 2 x is 2
not approaching a single real number, the limit of f (x) as x approaches 0 does not exist.
TABLE 11
x approaches 0 x approaches 0
2
from the left
−−−−−−−−−−−−−−−→
from the right
←−−−−−−−−−−−−−−−−
x −0.01 −0.001 −0.0001 → 0 ← 0.0001 0.001 0.01
f (x) = 1 10,000 10 6 10 8 f (x) becomes unbounded 10 8 10 6 10,000
4 2
2 4 x
x 2
x 0
y
4
2
2
4 2 4
4
Figure 49 f (x) = 1 x
2
x
However, since the numbers are increasing without bound, we describe the behavior of
the function near zero by writing
1
lim
x→0 x =∞ 2
and say that f (x) = 1 has an infinite limit as x approaches 0.
x
2
In other words, a function f has an infinite limit at c if f is defined everywhere in
an open interval containing c, except possibly at c, and f (x) becomes unbounded when
x is sufficiently close to c. ∗
EXAMPLE 1
1 1
Investigating lim and lim
x→0 − x x→0 + x
Consider the function f (x) = 1 . Table 12 shows values of f for selected numbers x
x
that are close to 0 and Figure 49 shows the graph of f .
TABLE 12
x approaches 0 x approaches 0
from the left
−−−−−−−−−−−−−−−−→
from the right
←−−−−−−−−−−−−−−−−−
x −0.01 −0.001 −0.0001 → 0 ← 0.0001 0.001 0.01 0.1
f (x) = 1 x
−100 −1000 −10,000 f (x) becomes unbounded 10,000 1000 100 10
Here as x gets closer to 0 from the right, the value of f (x) = 1 can be made as
x
large as we please. That is, 1 becomes unbounded in the positive direction. So,
x
Similarly, the notation
1
lim
x→0 + x =∞
1
lim
x→0 − x = −∞
is used to indicate that 1 becomes unbounded in the negative direction as x approaches 0
x
from the left. ■
∗ A precise definition of an infinite limit is given in Section 1.6.
128
Chapter 1 • Limits and Continuity
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