Section 1.5 Limits at Infinity, Infinite Limits
77
Problem Set 1.4
lit !,,,hlc,ns 1—14, evaluate each limit.
un
I,
22. Prove statements 5 and 6 of Theorem A using Theorem
1.3A.
urn (1 COS 0
2.
—
+
.
COS t
Inn—
1 + sin I
4. urn
.—
.
6
—
2x
-
sin 30
tan 0
7. urn
.
8. tim
i—o
—
i—n
tim
l)—()
-
cos I
sin x
r_
sin I
2
—
cos
sin3O
1.
and thus obtain another proof that tim (sin 1)/I
--
20
tan 50
sin 20
I’(cos
7
0
area (OBP)
tan x
3x
x—’()
sinx
urn
23. From area (OBP)
area (sector OAP)
area (ABPQ) in Figure 4, show that
sin3t
y
I. sin 1)
P(cos
2
sin!)
t.
12.i-
II.i-
I
sin 3t
H. urn
+ 4t
o
14. urn
t sec I
i—’O
B
J-ic .0)
I, 0!-’
B
-
0—
O—O
I’nthlems 15—19, plot the functions ti(x), 1(x), andf(x). Then
ii
“:
these graphs along with the Squeeze Theorem to determine
1(x).
IS. u(x)
16.
=
u(x) =
17. u(x)
=
18. u(x)
19. u(x)
=
lxi, 1(x)
xi, 1(x)
x
.1(x)
=
=
=
1,1(x)
=
I
2,1(x)
=
2
—lxi, f(x)
—lxi, f(x)
—
—
—
=
x sin(1/x)
=
x sin(i/x
)
2
f(x)
=
(1
, f(x)
2
x
=
2x
cos
, f(x)
2
x
=
+
x
,
—
cos
-,
x)/x
20. Prove that lim cos t = cos c using an argument similar to
I)1I one used in the proof that tim sin t = sin c.
21. Prove statements 3 and4 of Theorem A using Theorem
I IA.
1.5
Limits at Infinity
Infinite Limits
I +x
2
Figure 4
24. In Figure 5, let D be the area of triangle ABP and E the
area of the shaded region.
1)
(a) Guess the value of lirn
by looking at the figure.
-
(b) Find a formula for DIE in terms oft.
(c) Use a
Figure 1
I
I
I
I
I
2
.1
calculator to get an accurate estimate of
Answers to Concepts Review: 1. 0 2. I
nator is zero when t = 0 4. 1
lim
t—O
Li
3. the denomi
The deepest problems and most profound paradoxes of mathematics are often
intertwined with the use of the concept of the infinite. Yet mathematical progress
can in part he measured in terms of our understanding the concept of infinity. We
have already used the symbols 00 and —Do in our notation for certain intervals.
Thus, (3, Do) is our way of denoting the set of all real numbers greater than 3. Note
that we have never referred to cc as a number. For example, we have never added
it to a number or divided it by a number. We will use the symbols cc and —Do in a
new way in this section, but they will still not represent numbers.
Limits at Infinity Consider the function g(x) = x/(l + 2
x whose graph is
)
shown in Figure 1. We ask this question: What happens to g(x) as x gets larger and
larger? In symbols, we ask for the value of lim g(x).
Do, we are not implying that somewhere far, far to the
When we write x
right on the x-axis there is a number—bigger than all other numbers—that x is
Do as a shorthand way of saying that x gets larger
approaching. Rather, we use x
and larger without bound.
In the table in Figure 2, we have listed values of g(x)
) for several
2
x/(1 + x
values of x. It appears that g(x) gets smaller and smaller as x gets larger and larger.
—
I
Figure 5
We write