1 sided limits and continuity answers

CALCULUS BC
Name
HW4
, , )
\/{
-=T_T
on problems I - 4, sketch the graph ofa function that satisfies the
stated conditions.
/
l. / has a limit at-r- 3, but it is not continuous aix=
3.
2.
/
is not continuous at
to /(:)
3..f
r:
3, but
ifits
value at
= 0, it becomes continuous at
r:
¡-:
3 is changed from
/(3)
=
I
I
3.
has a removable disconrinuity at -r = c for which
/( c) is undefined. )-J
--++-JÍlc
4..f has a removable discontinuity at.r = c for which
/(c) is Oefined. ''"
On problems t5 - t,
7, Find
t,nd where each of these fi.rnc
continuity to prove the discontinuity at that value.
5. J\t
f (xt=t:t-:-o
(x=qvxr)
x-l
6. hþc)=
x-¡ @
x<0
!e'ir
l^
lx'if .r>0
d@
On problems 8 t t, use the
fu nction continuous everywhere.
-
i .L-
-'---W
-yL
À
It_4sin,
pþ)=1 x '
Ir-2,-s,
that will make the
k-5:l
.r<0
x>0
On problems 9-10, a function
7 -a u
conditions
ofthe Intermediate value Theo¡em hold fo;the give[value of À.
Ifthe conditions hold,
find a number c such that f (")= t Ifthe theo¡em does nÀt
hord, gìve the reason. whethe¡ the
theorem holds or not, sketch the curve and the lìne y :
¡.
.
9.
f
(x)=z¡¡-rz
Con{
o"r
fq'5]
À 'o(ø1" í4
.{-_-::l
¡,
,
.L
r'^fl
I;Él
--r+{-x'
-.,
i4
k-r
- 5-_
i,Sl
¡", i1=¡0,t1
frc*J'
x¿-1- \' Ð
;-
L:J
tl
fl¡)= I C1n+ o"r L3'
t (i¡-\t3
I
fl¡ì'
t"," ,),=u,'r'
- i^
5Å-,";
\-t-=:
'"y,_r;
lo.
t'o
,t, L
6
¡v :
3,
11' rn^1996 the united states postage rate fo¡ a first class
letter was $0.32 for the fìrsiilce and
$0.23 per ounce thereafter. Does the function meet the hypotheses
of the Intemediate value
Theorem? Is there a weight of rette¡ that can be mailed foi exactly
answers.
C_.3Â+,A3X
Crr'rl- .ñr o-íl K
$
1
.00? Justify your
,
if,
+,
J3Y:
t,
J
12. Jesse and Kay ran a 1000-m race. One minure affer
the
29 ko,/Llr , and Kay was running 1 5 km/hr. Three
minutes after lhe iace beguo,
f;,
slowed to lTkrn/hr,*ã ruv rru¿ speeded up to i9 kmfu.
Assume that each runner,s ,T""¿'
'
::::Tg3l::r,:-^ 1krj
r-.;
,,îî
"
lr
is a continuous function of time. prove thai there is a time
between 1 minute unJ¡
after-the ¡ace began at which each o¡e was ruming
exactry the sam. rp"eu. r, it p"r.iîr"
.-r1l,tY:*at+çd is? Is it possible ro rell whe-n thar sieed o.uo.d? e*pl^i,i. -Jyì,1
.i*t",
,(rìàoì
,
(5ìr r/
-3
C-
-,\
v-)Ò=-J(t-i\
n,,-"
*r,n
1¡,rbl
- '
(g,rcl
q/r
,,
s
'
]
= -31tV+ èt.f
,"1r-rt, -li+41
-7t
r
,
One-Sided Limits; Limits of Piecewise Functions
In Exercises l-3, use the graph to determine the limit, if it exists. Ifthe limit does not exist, explain
(u)
lyl(,)
(b)
i
4)
(")
lt$/(')
l'ïl(")
oll
|
b)t
b)À
(3. r)
c)
c=3
if it exists.
lim
/(x),
Ifit
wherc¡(x)=
does not ex
{.l
^
I t¿-zx
lim/{.r), where /{.r)
;:iJ-"'"'-'-rw
: l-4¡*6,_2, r<2
jg/k),
:
wtrere
/(r)
t>2 J
l_Ìr+4n
.I3+
l, r<l
r+1. Ì>
x,
I
1
.r< I
-r, ¡>
1
x'-4
y
-,)
tl-l
ri,n
¡*r- r'+f -+
,/L-1+
r"l"r('),
[;"o.Y
ir".
o
whereT(x)=le, ifx=0 DV{
[x'+
3,r'.
ifx
>o
¿
e.

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