cosx
5
-
-
0
—1
0
21(3n)
21(4it)
0
21(7it)
2/(8
Figure 8
‘2
0
0
-L
.1
it)
—1
2/(Ilit)
2/(12
0
2/ ( 10 it)
21(9it)
0
—1
2l(6it)
2/(5t)
2
sin-f
21(2 it)
2/it
x
Figure 7
2-
3-
Figure 6
0
3
•—o
0.000000005
± 0.01
.1.
0.2499 1
0.00990
0.5
± 0.1
s
2
0.99995
x
Chapter 1 Limits
± 1
58
4
—
COSX
1
io,oooj
02_
1
10,000
but to the left of c then
f(s) is near L.
f(x)
L means that when x is near
L means that when x is near but to the right of c then
f(.t) is near L. Similarly, to say that urn
=
Right- and Left-Hand Limits
To say that 1iin f(r)
Definition
One-Sided Limits When a function takes a jump (as does
at each integer
in Example 5), then the limit does not exist at the jump points. Such functions sug
gest the introduction of one-sided limits. Let the symbol x —. c mean that x ap
proaches c from the right, and let x —k c mean that x approaches c from the left.
Figure 9
y
SOLUTION This example poses the most subtle limit question asked yet. Since
we do not want to make too big a story out of it, we ask you to do two things. First,
pick a sequence of x-values approaching 0. Use your calculator to evaluate
sin (1/x) at these x’s. Unless you happen on some very lucky choices, your values
will oscillate wildly.
sin( l/x). No one will ever do this very
Second, consider trying to graph y
well, but the table of values in Figure 8 gives a good clue about what is happening.
In any neighborhood of the origin, the graph wiggles up and down between —1 and
1 infinitely many times (Figure 9). Clearly, sin (1/x) is not near a single number L
whenx is near 0. We conclude that lins sin(1/x) does not exist.
• EXAMPJ (Too many wiggles) Find lim sin(1/x).
check back, you will see that we have not claimed that every limit we can write
must exist.
.
1
10,000
SOLUTION Recall that [xli denotes the greatest integer less than or equal to x
is shown in Figure 7. For all numbers x less
(see Section 0.5). The graph of y =
= 2.
than 2 but near 2, jx] = 1. but for all numbers x greater than 2 but near 2,
Is j[xli near a single number L when x is near 2? No. No matter what number we
propose for L, there will be x’s arbitrarily close to 2 on one side or the other, where
I{Xli differs from L by at least Our conclusion is that limI[x] does not exist. If you
• EXAMPLE sj (No Iiniit at a jump) Find 1imx]j.
2
1
jim F
.—o[
SOLUTION Following the procedure used in Example 3, we construct the table
of values shown in Figure 6.The conclusion it suggests is that the desired limit isO.
But this is wrong. If we recall the graph of y = cos x, we realize that cos x
approaches 1 as x approaches 0. Thus,
• EXAMPLE (Your calculator niay fool you.) Find lim[x2
=
.
=
2
(x
2x
1)
5)
+
—
—
—
1)
when x
iirrif(x)=2
Figure 10
6. lirn(t
2
—
-4-
2t
2t)
—
_.
—
urn (x
2
2. lim (1
4.
x
—
—
2
4
r—’2
13. lim
x-.-1
11. lim
9. lim
(3t
+
5 -1- t
—
-
6)2
4)(t
—t2
x
x+1
2)
—
3
.x
+
2
x+6
4x
x—’2
7. lim
1)
x
—
2
9
x
3
—
-)
X
-
V(t 7)3
r—.T
r—7
14. lim
x—3
12. urn
)
4
r—’
10. lim
t+7
-t-4t—21
2
t
-l-2x
4
x
—
3
2
x
i——7
8. urn
and
Lif and onlyif iimf(x)
In Problems 7—18, find the indicated limit. In most cases, it will be
wise to do some algebra first (see Example 2).
.
x—’—2
3. lim
x—3
1. lim(x
In Problems 1—6,find the indicated limit.
Problem Set 1.1
—
1
urn If xl
x—’2
2
=
Land 1imf(x)
L.
V
—
h
—
4
+
3)2
2
18x
(2 + h)
2
(x
—
81
u—’l
18. lim
16. jim
h
(x + h)
2
—
—
—
1)2
4)(2u
(u
(3it +
2)
(x
1
+
1)
—
X
—
—
sin(x
i
32r/2)
1)2
—
sin x)
2
—
—
—
(x
x—it/4 (tan x
lim
25. jim
27.
sinx
2x
—
i—.lsifl(t
23. lim
x—O
21. urn
x—O
19. urn
28.
•
lim
1
x
—
2
2
—
1/t
2 sin u
3u
cott
x—3
—
2
cosx)
sin(x
2t
—
—
(1
u—ir/2
r—O
26 lim
24. lim
x—()
22. urn
i—O
20. lim
_—
1
3)
—
3
In Problems 19—28, use a calculator to find the indicated limit.
Use a graphing calculator to plot the function near the limit point.
h—O
17. urn
x—’3
15. limX
-
1im f(s) = L means thatf(x) gets near to
when x
approaches c from the
4. If both lim f(x) = M and lim f(x) = M, then
)=4
3.
Iirnf(x
Figure 10 should give additional insight. Two of the limits do not exist, although all
but one of the one-sided limits exist.
Theorem A
L means that f(x) gets close to
—
lim{xl
x—’2
We believe that you will find the following theorem quite reasonable.
gets sufficiently close to (but is different from)
2. Let f(s) = (2 9)/(x 3) and note thatf(3) is undefined. Nevertheless, hinf(x) =
1. lim f(x)
Concepts Review
x—2
59
Thus, while lim [xl does not exist, it is correct to write (look at the graph in Figure 7)
Section 1.1 Introduction to Limits
60
Chapter 1 Limits
29. For the function f graphed in Figure 11, find the indicated
limit or function value, or state that it does not exist.
(c) f(—1)
(b) f(—3)
(a) lirnf(x)
(d)
lim f(s)
(e) f(l)
(g)
limf(x)
(h)
2
37. Find lim(x
limf(x)
(i)
(d)
limf(x)
lim f(x)
2
1
x—’1
36. Follow the directions of Problem 35 forf(x)
liin f(s)
(f)
limf(x)
(c)
1)/x
—
38. Evaluate Iim(
=
x/IxL
or state that it does not exist.
—
—
\/)/x. Hint: Rationalize the
numerator by multiplying the numerator and denominator by
V
3-
y
{
39. Let
/.<:
I.
—3
I
I
—2
—I
-4
—3
-2
f(x)
—
t
—l
Figurel2
Figurell
30. Follow the directions of Problem 29 for the function
graphed in Figure 12.
f
31. For the function f graphed in Figure 13, find tue indicated
limit or function value, or state that it does not exist.
(c) lirn f(x)
(a) f(—3)
(b) f(3)
limf(x)
0-
1(x)
(e)
lirnf(x)
=
(
—3 —2 —I_i
Th
I
2
i
\I
4X
I
3
4
5
X
•
f(x)
l*f(x)
=
f that
1
41. Let
2
1x
=
For what values of a does urn
I
2
x—’O
2
f(x)
y
(b) lim
40. Sketch, as best you can, the graph of a function
satisfies all the following conditions.
(a) Its domain is the interval [0, 4].
(b) f(0) = f(1) = f(2) = f(3) = f(4) = 1
(d) urn f(x) = 1
(c) lirnf(x) = 2
limf(x)
(f)
\\
—
2
-4
urn.
(e)
ifxisrational
if x is irrational
Find each value, if possible.
(a) lim f(x)
.x- -3
(d)
x
—x
if x is rational
if x is irrational
f(s)
exist?
42. The function f(x) = x
2 had been carefully graphed, but
during the night a mysterious visitor changed the values off at a
million different places. Does this affect the value of lim f(x) at
any a? Explain.
—‘3
—3
-4
-
Figurel4
Figurel3
43. Find each of the following limits or state that it does not
exist.
Ix ii
Ix— 1
(b) lim
(a) lim
.t—’1 x
x—’l x
1
1
1
1_
Ix
ii
11
(d) lim
(c) urn
xi- x—1
•—‘1—
Is—il]
Ix ii
—
—
—
32. For the function f graphed in Figure 14, find the indicated
limit or function value, or state that it does not exist.
(c) lirn f(x)
(b) lirnf(x)
(a) liinf(x)
(d)
f(—i)
(e) Iimf(x)
(f) f(1)
33. Sketch the graph of
(
—x
ifx<0
ifOx<1
ifx1
Then find each of the following or state that it does not exist.
(a) urn f(s)
(b) liirif(x)
(c) f(1)
(d)
urn
i—I
f(x)
34. Sketch the graph of
—+1
x—l
52
ifx<1
ifl<x<2
jfx>2
Then find each of the following or state that it does not exist.
(a) lint g(x)
(b) g(1)
(c)
urn
x-2
g(x)
—
[
—
—
44. Find each of the following limits or state that it does not
exist.
[xJ
(b) lim[1/x]
(a) lim\/x
—
f(x)=
x
L.1-i-x
g(x)=
—
(d)
limg(x)
35. Sketch the graph of f(.x) = x
x; then find each of
the following or state that it does not exist.
(b) limf(x)
(a) f(O)
—
(c)
lim x(—1)
1
/11
(d)
lim[x](_1)Lh/xl
45. Find each of the following limits or state that it does not
exist.
(b) lim. x
[1/x]
2
(a) lim xj1/x
x—O
(c)
lim_(xj
+
[—xfl
(d)
lim+([xl
+
[—x])
4. Find each of the following limits or state that it does not
exist.
(a) lim[r]/x
(b) lim[x]/x
x—O
(c)
Lim [x
(d)
lim [xI/x
x—’1.8
Many software packages have programs for calculating lim
its, although you should be warned that they are not infallible. To
develop confidence in your program, use it to recalculate some of
the limits in Problems 1—28. Then for each of the following, find
the limit or state that it does not exist.
47. urn
48. lim xX
49. urn
x—0
50. lim
xIx