1. Exercises 2.4 87
For the function defined by (9). both parts (i) and (iii) of the above definition
are true. See Figure 4. If 9 is the function defined by
1
g(x) = - (x _ a)2
then both parts (ii) and (iv) are true, and the line x = a is a vertical asymptote
of the graph of g. This is shown in Figure 9.
y
EXA.MPLE 4 Find ihe vertical asymptote and draw a sketch of the graph
of the function defined by
3
f(x) =--3
x-
Solution
3 ' . 3
Jim --'= +:::1) I
1m -- = -00
x-3+ X - 3 x-3- x-3
It follows from Definition 2.4.8 that the line x = 3 is a verticai asymptote of
FIGURE 10 the graph of f. A sketch of the graph of f appears in Figure 10.
EXERCISES 2.4
In Exercises I through J 2, do the following: (a ) f.!.se a calculator x+2
6. (a) fix) = ---2; x is 0, 0.5, 0.9, 0.99, 0.999, 0.9999 and x is
to tabulate values of f(x) for the specified values of x, and from lx-I)
these values make a statement regarding the apparent behallior of . x+2
f(x). (b) Find the indicated limit. 2, 1.5, 1.1, 1.O 1. 1.00 I, I. 000 I; (b) lirn --2
x-I (x - 1)
x - 2 .
1. (a) f(x) = x _ 5; x IS 6, 5.5, 5.1, 5.01, 5.001, 5.000 I;
I . " 7. (a) fIx) = --; x IS 0, -0.5. -0.9, -0.99, -0.999,
x + I
. I x-2
(b) x - 5
-0.9999; (b) lim --
ie
x~'
x--I' x + I
5, I x-2
2. (a) fix) = ;=5; x is 4, 4.5, 4.9, 4.99, 4.999, 4.9999; 8. (a) fix) = x + I; x is -2, -1.5, -1.1, - 1.01, -1.001,
7
~ I x-2
(b) }~~_ x _ 5 - 1.0001;(b) lim --
%-·,1- X +!
I .
3. (a) f(x) = (x _ w; x IS 6, 5.5, 5.1, 5.01, 5.001,5.0001 and x 9. (a) fix) = _x_; x is -5,
x+4
-4.5, -4.1, -4.01, -4.001,
oX I
is 4, 4.5, 4.9, 4.99, 4.999, 4.9999; (b) lim ---2 -4.0001; (b) lim __x_
.-5(x-5) x- -4- X +4
x+2 . x .
4. (a) fix) = 1 _ x; x IS 0, 0.5, 0.9, 0.99. 0.999, 0.9999; 10. (a) fix) = --4;x IS 5. 4.5, 4.1, 4.01, 4.001, 4.0001;
x-
x+2
(b) urn -1- lim _x_
(b)
x ...•l " x x-4·x-4
x+2 . 4x
5. (a) fix) = 1 _ x; x IS 2,1.5,1.1.1.01,1001,10001; 11. (a) fIx) = ---2; x is -4, - 3.5, - 3.1, - 3.01, - 3.001,
9 - X'
x+2 4x
(b) lim -I - -3.0001; (b) lim --2
$-1" - X x--3- 9- x

2. -
88 limits and Continuity
I
4X2 1 J 1
12. (a) fix) = --2;
9-x
X is 4, 3.5, 3.1, 3.01, 3.001, 3.0001;
.
graph: (a) Jx) =;; (b) g(x) = ?; (c) hex) = 0;
4X2 . I
(b) lim --2 (d) <p(x) = 4'
'x_"+9-x x
34. For each of the following functions, find ill, verticalasyrnp,
In Exercises 13 through 32, find the limit.
tote of the graph of tile function, and draw a sketch of
t +2 -1+ 2 1 I I
1}. lim -2-- 14. lim (-2)2 the graph: (a) J(x) = --; ~b)(x)
g = -2; (c) h(x) = -3;
I-pt -4 1-2- r- x x x
J
1+
2 '3 + X2 (d) <p(x) = -4'
is: I'rm
. -2-- 16. lim _"_._
x
1-2-t -4 x-o- x
17. lim ,)3 + X2 18. lim.j) + X2 In Exercises 35 through 42, find the verrical asympore(s) oj the
.x-o- X x-v .,!2 graph oj the Junction, and draw a sketch of the graph.
19. lim
,)x2-9
20. lim v
,~
- x- ., 3
x-p X- 3 x-4- X- 4 35. f(x)=~4 36.(x) =-1
f
x+
2!. lim - -
. (1 1) 2 22 lim
xl -
-3--2
3 -2
37. fix) = --3
x-
38. fix) = --5
-4
x-O' ~ X r-O-X +x x+ x-
3
23 lim 2 - 4x 24. lim (_1 3_) 4
. x-o- 5r + 3r <-2- ,s - 2 S2 -'4 39. J(x) = (-3
x +
-2
2
) 40. fix) = (x _ 5)2
2S Jim ( 2 __ 3) . 5 1
• 1_ -4- 12 + 3r - 4 t +4 41. fix) = -X""'2-+-"8-x-+-1-:-5 42. fix) = 2 6
x + 5x-
2x3 _ 5x2
26. lim -.----:--
x-l- X2 - 1 43. Prove that lim __ 3_2 = + CXJ by using Definition 2.4.1.
[X2] - 1 x-2(x-2)
21. lim [xl - x 28. lim ---
x-3- 3-x x-,- X2 - I
44. Prove that lim ~ = - CXJ by using Definition 2.4.2.
x-4 (x - 4)
x3 + 9x2 + 20x 6x2+-x-2
29. lim -..,,----:_=__ 30. lim .,....-,2:-----
45. Prove Theorem 2.4.3(ii). 46. Prove Theorem 2.4.4(ii).
x-3 x2+x-12 x--2·2x +3x-2
47. Prove Theorem 2.4.4(iii). 48. Prove Theorem 2.4.4(iv).
x-I x-2
31. lim ---=- 32. lirn --' -===
x-~- 2 - ,)4x - X2
49. Prove Theorem 2.4.5. 50. Prove Theorem 2.4.6.
x-,·..)2x-x2-1 51. Prove Theorem 2.4.7.
33. For each of the following fun tions. find the vertical asymp- 52. Use Definition ,
2.4.1 to prove that lim -- 15 - xl = -i- 00.
tote of the graph of the function. and draw a sketch of the <--33+y
2.5 LIMiTS AT INFINITY The previous section was devoted ,0
infinite limits where function values either
increased or decreased without bound as the independent variable approached
a real number. Wc now consider limits of functions when the independent
Table 1 variable either increases or decreases without bound. We begin with the func-
---!-·-----·.... -~.2
tion defined by
x fIx) = -;~--
x- + 1
IY:
o (j
J(x)=-~-
1 1 y2 + I
2 1.6
3 1.8
1.882353 Let x take on the values O. I. 2.. -. -+.5. 10. 100. 1000, and so on, allowing x to
4 I
5 1.923077 increase without hound. The eorr.:,;pontiing function values. either exact or ap-
10 1.9XOI'lX proximated by a calculator to six decimal places, are given in Table 1. Observe
100 1.9l)'!~O()
from the table that as .v inCl'ca,cs through positive values, the function values
I~ __ -
1.'!999l}X
get closer and closer to 2.

3. bwcisa2.5 97
ominator by by Definition 2.4.8(1)the line x = 2 is a vertical asymptote of the graph of fl'
e have
lim [,(x) ~ lim 2
%-+00 %-+00
J 1 __
I 2
x
=2
Thus by Definition 2.5.4(i)the line y = 2 is a horizontal asymptote of the graph
offt.
Similarly, lim flex) = 2 A sketch of the graph of fl is in Figure 9.:
10
11
I
!~
= -00
Hence, by Definition 2.4.8(ii)the line x = 2 is a vertical asymptote of the graph
? Ii"
of-f2' .
ntal asymptote.
. ----·2
10
----1--------
2 I"
lim f2(x}
%-+co
= lim
.x-+oo
[-2) 1 __
I 2]
x
_____ -2~--1-------
~v
= -2
d draw a sketch
Thus, by Definition 2.5.4(i}the line y = - 2 is a horizontai asymptote of the
graph of f2'
Also, lim f2(x) = - 2. A sketch of the graph of f2 appears in Figure 10.
11 The graph of the given equation is the union of the graphs of fl and f2' and
a sketch is in Figure 11.
ISES 2.5
EJocises 1 through 10, do the following: Use a calculator to
zie the values of f(x) for the specified values of x. (a) What
S. f(x) = -
x +1
;X2 ;
x is 0, 1, 2, 4, 6, 8, !O, 100, 1000and x is
.
:-'x) 'appear /0 be approaching as x increases without bound? -1, -2, -4, -6, -8, -to, -100, -1000.
doesf(x) appear to be approaching as x decreases with·
x3
. (c) Find Jim f(x). (d) Find lim f(x), 6. f(x) = -3--; x is 2, 4, 6, 8,10,100,1000 and x is -2, -4,
x- +«1 x +2
-6, -8, -10, -100, -1000.
x) = ~2; x is 1,2,4,6,8,10,100, toOO and x is -1, -2, 4x + 1
- x 7. f(x) = --; x is 2, 6, to, 100, 1000, 10,000,100,000and
2x - 1 .
_j. -6, -8, -10, -100. -1(01).
x is - 2, - 6, - 10, - 100, - 1000, - to,OOO, - 100,000.
e two functions 5x - 3
values of x for _ xl =~; x is l, 2, 4, 6, 8, 10,.100,1000and x is -1. -2, 8. f(x) = --; x is 2, 6,10, 100, 1000, to,OOC, 100,000and
x lOx +1 .
'on 1.5 and ex- _.!. -6, -8, -10, -100, -1000. x is -2, -6, -to, -100, -1000, -10,000, -100,000.
~u (2, + (0).
'-xl = ~; x is 1,2,4,6,8, 10, 100, 1000and x is -I, -2, 9. f(x) = x'~ 1; x is 2, 6, 11), 00,1000,10,000,100,000
1 and x is
x x
_.!. -6. -8, -lO, -100, -1000. -2, -6, -Hi, -100, -1000, -10,000, -100,000.
?
) = -.;; x is 1,2,4,6,8, to, 100,IC'.lO
and x is -1, -2, 10. f(x) = ~; x is 2, 6, 10,100,1000,10,000,100,000and x is
x x+l
_!. - 6. - 8, -10, -100, -1000. -2, -6, ..cl0, -100, -1000, -10,000, -100,000.

4. limits and Continuity
In Exercises 11 through 30, find the limit. x'
44. C/(x) = -'-
. 4 - x'
21 -e- j 6x - 4
II. Jim .-- 12. lirn 2x . -
, __ o 5t - 2 X-4 '":7 3x + 1 45. hl.') = ------ 46. fIx) = r=;;====
2x + 7 i + 5x
6x' + llx - 10 Jx' + 5x + 6
13. Jim -;:-- 14. lim ---:;--
x
x-« -" 4 - ox x_+.:c,2-.Jx
48. h(x)=~
7x2 - 2x + 1 · 4S2 +3 ... -
':x' 9
15. lim ---02,.----- 16. hm -.--
x-+003x +8x+5 ,--oc 2s' - 1
In Exercises 49 through 56. find the horizontal and vertical as-
x+4 x' + 5
17. lim -2-- 18. lim --J- ymptotes and draw a sketch of the graph of the equation.
x_Tx.3x - S .1'-. 'S' X
49. 3xy - 2x - 4y - 3 = 0 SO. 2xy + 4x - 3y + 6 = 0
lv' - 3y · x' - 2x + 5
19. lim -'--- 20. lim -;);---- 51. x'y' - x' + 4y' = 0 52. xy' + 3y' - 9x = 0
y_+'" y+ 1 ,_.OX) 7.>: + x + 1 54. 2xy' + 4y' - 3x = 0
53. (y' - l)(x·- 3) = 6
4x3 + 2x' - 5 · 3x4 - 7x' + 2 55. x' y .. 2x' - y - 2 = 0
21. Jim -3----- 22. lirn -----.
x--'" 8x + x + 2 x-+ x 2x· + 1 56. x'y + 4xy - x' + x + 4y - 6 = 0
2v3 - 4 5x3 - 12x +7
23. Jim -'--- 24. lim 4x' _ 1 In Exercises 57 through 60, prove that Jim f(x) = 1 by applying
y_+",Sy+3 Z-+a;)
Definition 2.5.1; that is, for any € > 0 show that there exists a
25. x~~'" (3X + :,) 26. lim (~-
, ....•-+.x. (
4t) number N > 0 such that ifx > N. then If(x) - 11 < e.
. ,,/x' +4 JX2 +4 x 2x
27. 11m --- 28. lim --- 57. fIx) = x _ 58. f(x) = 2x +3
,r-04 T: X +4 x--x x+4
,j'''''''''---=2-w-,-' -=-3 . Jy. + 1 x' - 1 X2 + 2x
29. lim ------ 30. I trn --- 59. f(x) = X2 +1 60. f(x) = ~
w+S v __ co 2/ - 3
In Exercises 31 through 36, find the limit (Pint: First obtain a . 8x +3 .
[racti.n. ii;; .1rational numerat or.} 61. Prove that 11m -- = 4 by showing that for any" >0
x--0:2-,,-
31. 111.1 (,. ::: + 1- x] there exists a number N < 0 such that if x < N then
,r ~ t .., .'(- .•.. ~ 8.>: +...2 - 41 < E.
34. hrn I/~~ I - x) 1 2x - I
x-" r: r- ----;:;::=..:....=
62. Prove part (i) of Limit Theorem 2 (2:4.4) if "--; ....•a" is re-
36. lirn ,./ t +..:! t + ..,fr
placed by "x ....•+ "f."
,-.'" .../1+1 63. Give a definition for each of the fellowing:
In Exercises 37 through 48. find the horizontal and vertical as- (a) lim f(x) = - x; (b) lim f(x) = + 00;
x ....•+0:
ymptotes and draw a sketch of the graph of the function. (c) Jim f(x) = -"X:.
70-- <T.
2x + I 4 - 3x 64. Prove that lim (x' - 4) = + 00 by showing that for any
37. fIx) = --1 38. {(x) =--
x'· , . .x + I
N > 0 there exists an M > 0 such that if x > M then
.
39. g(x) = 1 . - 40. "(x) = : + "2 x' - 4> N.
x x 65. Prove that lim 16 - x - x') = - 00 by applying the de-
x-+a:
2
41. f(x) = ,- (.')=-'= 3x
41 . Ftx' - finition in Exercise 63(a).
,.,··4 ,'x' +3 66. Prove part (ii) of Limit Theorem 13 (2.5.3).
2.6 CONTINUITY OF A In Illustration I of Section 2.3 we discussed the function C de/hied by
FUIVCTIOIt AT A NUMilER
C(X)={x ifO:s;x:s;lO (1)
0.9x iflO<x
where C(x) dollars is the total cost of x pounds of a product. We showed that
Jim C(x) does not exist because Jim C(x)"# Jim C(x). A sketch of the graph
70-10 x-lO- x-to.