xlim!x0
f .x/DL1 and lim
x!x0
g.x/DL2; (9)
then
xlim!x0
.f Cg/.x/DL1CL2; (10)
xlim!x0
.f g/.x/DL1 L2; (11)
xlim!x0
.fg/.x/DL1L2; (12)
and, ifL2 ¤0, (13)
xlim!x0
f g
.x/D L1
L2
: (14)
Proof
From (9) and Definition 2.1.2, if > 0, there is aı1> 0such thatjf .x/ L1j< (15)
if0 <jx x0j< ı1, and aı2> 0such that
jg.x/ L2j< (16)
if0 <jx x0j< ı2. Suppose that
0 <jx x0j< ıDmin.ı1; ı2/; (17) so that (15) and (16) both hold. Then
j.f ˙g/.x/ .L1˙L2/j D j.f .x/ L1/˙.g.x/ L2/j jf .x/ L1j C jg.x/ L2j< 2;
which proves (10) and (11).
To prove (12), we assume (17) and write
j.fg/.x/ L1L2j D jf .x/g.x/ L1L2j
D jf .x/.g.x/ L2/CL2.f .x/ L1/j jf .x/jjg.x/ L2j C jL2jjf .x/ L1j .jf .x/j C jL2j/ (from (15) and (16)) .jf .x/ L1j C jL1j C jL2j/
.C jL1j C jL2j/ from (15) .1C jL1j C jL2j/
if < 1andxsatisfies (17). This proves (12).
To prove (14), we first observe that ifL2¤0, there is aı3> 0such that jg.x/ L2j< jL2j
2 ; so
jg.x/j> jL2j
2 (18)
if
0 <jx x0j< ı3:
To see this, letLDL2andD jL2j=2in (4). Now suppose that 0 <jx x0j<min.ı1; ı2; ı3/;
so that (15), (16), and (18) all hold. Then
ˇˇ
Successive applications of the various parts of Theorem 2.1.4 permit us to find limits without the–ıarguments required by Definition 2.1.2.
Example 2.1.7
Use Theorem 2.1.4 to findxlim!2
9 x2
xC1 and lim
x!2.9 x2/.xC1/:
Solution
Ifcis a constant, then limx!x0cDc, and, from Example 2.1.5, limx!x0xD x0. Therefore, from Theorem 2.1.4,xlim!2.9 x2/D lim
satisfies the inequality
jf .x/j<
if0 < x < ı D =2. However, this does not mean that limx!0f .x/ D 0, sincef is not defined for negativex, as it must be to satisfy the conditions of Definition 2.1.2 with x0D0andLD0. The function
g.x/DxC jxj
x ; x¤0;
can be rewritten as
g.x/D
xC1; x > 0;
x 1; x < 0I
hence, every open interval containingx0 D 0 also contains pointsx1 andx2 such that jg.x1/ g.x2/j is as close to 2as we please. Therefore, limx!x0g.x/does not exist (Exercise 26).
Althoughf .x/andg.x/do not approach limits asxapproaches zero, they each exhibit a definite sort of limiting behavior for small positive values ofx, as doesg.x/for small negative values ofx. The kind of behavior we have in mind is defined precisely as follows.
y
x0 x
x x0 − x x0 +
f(x) = λ
y = f(x)
f(x) = µ
lim lim
µ λ
Figure 2.1.2 Definition 2.1.5
(a)
We say thatf .x/approaches the left-hand limitLasxapproachesx0from the left, and writex!limx0
f .x/DL;
iff is defined on some open interval.a; x0/and, for each > 0, there is aı > 0 such that
jf .x/ Lj< if x0 ı < x < x0:
(b)
We say thatf .x/approaches the right-hand limitLasx approachesx0from the right, and writex!limx0Cf .x/DL;
iff is defined on some open interval.x0; b/and, for each > 0, there is aı > 0 such that
jf .x/ Lj< if x0< x < x0Cı:
Figure 2.1.2 shows the graph of a function that has distinct left- and right-hand limits at a pointx0.
Example 2.1.8
Letf .x/D x
jxj; x¤0:
Ifx < 0, thenf .x/D x=xD 1, so
xlim!0 f .x/D 1:
Ifx > 0, thenf .x/Dx=xD1, so
xlim!0Cf .x/D1:
Example 2.1.9
Letg.x/D xC jxj.1Cx/
x sin1
x; x ¤0:
Ifx < 0, then
g.x/D xsin1 x; so
xlim!0 g.x/D0;
since
jg.x/ 0j Dˇˇˇˇxsin1 x ˇˇˇ
ˇ jxj<
if < x < 0; that is, Definition 2.1.5
(a)
is satisfied withı D. Ifx > 0, then g.x/D.2Cx/sin1x;
which takes on every value between 2and2in every interval.0; ı/. Hence,g.x/does not approach a right-hand limit atxapproaches0from the right. This shows that a function may have a limit from one side at a point but fail to have a limit from the other side.
Example 2.1.10
We leave it to you to verify thatx!lim0C
jxj x Cx
D 1;
xlim!0
jxj x Cx
D 1;
xlim!0Cxsinp
xD 0;
and limx!0 sinp
xdoes not exist.
Left- and right-hand limits are also calledone-sided limits. We will often simplify the notation by writing
x!limx0
f .x/Df .x0 / and lim
x!x0Cf .x/Df .x0C/:
The following theorem states the connection between limits and one-sided limits. We leave the proof to you (Exercise 12).
Theorem 2.1.6
A functionf has a limit atx0if and only if it has left- and right-hand limits atx0;and they are equal. More specifically;xlim!x0
f .x/DL if and only if
f .x0C/Df .x0 /DL:
With only minor modifications of their proofs (replacing the inequality0 <jx x0j< ı byx0 ı < x < x0orx0 < x < x0Cı), it can be shown that the assertions of Theo-rems 2.1.3 and 2.1.4 remain valid if “limx!x0” is replaced by “limx!x0 ” or “limx!x0C” throughout (Exercise 13).
Limits at
˙1Limits and one-sided limits have to do with the behavior of a functionf near a limit point ofDf. It is equally reasonable to studyf for large positive values ofxifDf is unbounded above or for large negative values ofxifDf is unbounded below.
Definition 2.1.7
We say thatf .x/approaches the limitLasx approaches1, and writexlim!1f .x/DL;
iff is defined on an interval.a;1/and, for each > 0, there is a numbersuch that jf .x/ Lj< if x > :
Figure 2.1.3 provides an illustration of the situation described in Definition 2.1.7.
x ∞ lim f(x) = L
β y
L + L − L
x
Figure 2.1.3
We leave it to you to define the statement “limx! 1f .x/D L” (Exercise 14) and to show that Theorems 2.1.3 and 2.1.4 remain valid ifx0is replaced throughout by1or 1 (Exercise 16).
Example 2.1.11
Let f .x/D1 1x2; g.x/D 2jxj
1Cx; and h.x/Dsinx:
Then
xlim!1f .x/D1;
since
jf .x/ 1j D 1
x2 < if x > 1 p; and
xlim!1g.x/D2;
since
jg.x/ 2j D ˇˇ ˇˇ 2x
1Cx 2 ˇˇ ˇˇD 2
1Cx < 2
x < if x > 2 :
However, limx!1h.x/does not exist, sincehassumes all values between 1and1in any semi-infinite interval.;1/.
We leave it to you to show that limx! 1f .x/ D 1, limx! 1g.x/ D 2, and limx! 1h.x/does not exist (Exercise 17).
We will sometimes denote limx!1f .x/and limx! 1f .x/byf .1/ andf . 1/, respectively.
Infinite Limits
The functionsf .x/D 1
x; g.x/D 1
x2; p.x/Dsin1 x; and
q.x/D 1 x2sin1
x
do not have limits, or even one-sided limits, atx0D0. They fail to have limits in different ways:
f .x/increases beyond bound asxapproaches0from the right and decreases beyond bound asxapproaches0from the left;
g.x/increases beyond bound asxapproaches zero;
p.x/oscillates with ever-increasing frequency asxapproaches zero;
q.x/oscillates with ever-increasing amplitude and frequency asxapproaches0.
The kind of behavior exhibited byf andg near x0 D 0 is sufficiently common and simple to lead us to defineinfinite limits.
Definition 2.1.8
We say thatf .x/approaches1asx approachesx0 from the left, and writex!limx0
f .x/D 1 or f .x0 /D 1;
iff is defined on an interval.a; x0/and, for each real numberM, there is aı > 0such that
f .x/ > M if x0 ı < x < x0:
Example 2.1.12
We leave it to you to define the other kinds of infinite limits (Exer-cises 19 and 21) and show thatxlim!0
1
x D 1; lim
x!0C
1 x D 1I
xlim!0
1
x2 D lim
x!0C
1 x2 D lim
x!0
1 x2 D 1I
xlim!1x2D lim
x! 1x2D 1I and
xlim!1x3 D 1; lim
x! 1x3D 1:
Throughout this book, “limx!x0f .x/exists” will mean that
xlim!x0
f .x/DL; whereLisfinite.
To leave open the possibility thatLD ˙1, we will say that
xlim!x0
f .x/ exists in the extended reals.
This convention also applies to one-sided limits and limits asxapproaches˙1.
We mentioned earlier that Theorems 2.1.3 and 2.1.4 remain valid if “limx!x0” is re-placed by “limx!x0 ” or “limx!x0C.” They are also valid withx0 replaced by ˙1. Moreover, the counterparts of (10), (11), and (12) in all these versions of Theorem 2.1.4 remain valid if either or both ofL1andL2are infinite, provided that their right sides are not indeterminate (Exercises 28 and 29). Equation (14) and its counterparts remain valid if L1=L2is not indeterminate andL2¤0(Exercise 30).
Example 2.1.13
Results like Theorem 2.1.4 yieldxlim!1sinhxD lim we cannot obtain limx!1f .x/by writing
xlim!1f .x/D lim
x!1e2x lim
x!1ex;
because this produces the indeterminate form1 1. However, by writing f .x/De2x.1 e x/;
Example 2.1.15
Letg.x/D 2x2 xC1 3x2C2x 1:
Trying to find limx!1g.x/by applying a version of Theorem 2.1.4 to this fraction as it is written leads to an indeterminate form (try it!). However, by rewriting it as
g.x/D 2 1=xC1=x2
3C2=x 1=x2; x¤0;
we find that
xlim!1g.x/D
xlim!12 lim
x!11=xC lim
x!11=x2
xlim!13C lim
x!12=x lim
x!11=x2 D 2 0C0 3C0 0 D 2
3: