Section 2.2 Exercises
5.
1.
I-2O,2OlttyJ-a,41
I-5,51by[-r.s, 1.5]
x
tt
13t
100
tr
ffi
ro0
.t0e
'200
.ta0
f tE (5x+l )/(ahs(.-
(a) lim f (x)=3
(a) limf (x)=l
(b) Itutrf (x)=-3
O) lim f (x)=l
(c)y=3,y=-3
(c) y= I
9. 0 < I - cos.r3 2. So,for x > 0 we have 0 ( I - cos.r-<4.
x"
By the SandwichTheorem,
l -cosr
.. I
0= lim(0)=lim:"+== hi=0.
x1@
l-t-
x'
t-5, sl by I-10, 101
11. -l 3si nr< l
j=
lim 1=0.
ti*
t14
:100 | -tEtrl
-100 | -tEt{
-300 | Eti[[
x
,-+-
x
13.
(a) lim f (x)=0
(b) fim /(x)=*
(c ) y = o
[-2,6l by[-r, 5]
4.
l/
t-10, l0l by [-100,300]
(a) lim /(.r) =
-
(b) fim /(r)=*
(c) No horizontalasymptotes.
I
x-2
lim -=€
r+t
x)a
x-
2il. Usethemethodof Examplel0 in thetext.
/t\
cosl - |
u m \ l/ = [ * * rr= c o s (o )_] = ,
ri -
[-7, U by [-3,3]
''.t
.tt
.tt
.3.4
9.1
.tl
.1.001
f i.
'1.t5
-e.5
't.lll
.E
-10
-100
-1000
r*
,,I
rf-
x+orl *x
l +0
1
"",ir)
\ i/ = [ - * r" = * r(o )= l=r
,l ?,- 1
r +o - l *x
l +0
I
-
atttl(x+.tt
n.
6g1r=oo
x-+-{ x*3
17.
t-4, 4lw [-3,3]
(a)x= -2,x= 2
(b) Left-handlimit at -2 is
-.
Right:handlimit at -2 is *.
Left-handlimit ar 2 is *.
Right-handlimit at 2 is
-.
[-4, a] by t-3, 3l
x
0
0
0
0
0
0
0
.N
.'|
.z
.t
.01
.001
tE't
x)/
29.
.. intx
llm - = u
x+0'
X
-l
t-6 el uyI-r2,61
(a)r= -l
(b) Left-hard limit at -l is *.
Right-handlimit at -l is
-.
l-3, 3l by t-3, 3l
x
.t
.t
.t
.l
.0t
.00t
lE-\
l.}rt
z.5t7t
9,0335
toot?
100
t00n
10000
35. An endbchaviormodElis t=2y2.
lr El/Ern (x)
lim cscr =
r+u
36. An endbehaviormodelis
-
*
n,=(z-t.)(#)
=(ryF)[
#)
=(ry\(
-t-)-
\ x + t / \ 51 1 2 /
.r3+z-r2
x 3+ x 2 + 5 x + 5
An endbehaviormodelfor y is
lim1 =1
'9'=
,9 Y=" 9 t=t
= t.
x-"-
1s,,)
= O.S*,. t"l
37. An endbehaviormodelis 3xt =
-2r1. 14;
3E.An endbehaviormodeti*
4
= -"r.
(U)
41.(a) -rl
z *'= 2 ,
(b)y= o
1- 3
43.@)7!--4tz
x
(b) None
45. (a) The function y = e' is a right end behavior model
because
r+e
e,
^e*-2x
= u-[t-4.)=t-o=1.
r+-\
e, )
(b) The function y = -21 is a left end behavior model
=
b ecause
x+*
-2x
^e'-2x
2x
^(-{,l=o*r
) = r.
'+*[
,fE. (a, b) The function y = 12 is both a right end Uefravio,
model and a left end behavior model because
lc
\
n, [ " ' * ! i n 'l =r+t* | . t * l r y ] = r .
r++-(
x.
,/
"\
x.
)