set 2: Murtipre-choice Questions
on Limits and continuity
Set 2: Multiple-Choice
Questions on Limitsand
Continuity
..t
I.
.i
I
.r- * 4
Iirn-l-I
.r-t
;*
(A) 1
G)o
(c) -'')
G)0
(c) 4
I
.r,.-
oo
I
(A) I
r.
J.
111n
,--..i
G)l
1
(c)
4
{ir1}
_1
(E)
oo
(D) oo
(E)
none of these
is
(A) r
.r'l - 8
.. ---,
lrr1l
.
r
(D)
.t-j
*--_-=--.i.
i- - f .t - -1 '-
(A)0
5.
(E)
-1
-1
'=- '.j' i.
linr
,_.- _
2.
,1.
(D)
-l .I-- -l
(B)
o
(c) F
(D) -t
(B)
0
(c) I
(D)
(E)
nonexistent
is
(A) 4
(E) €
3
J-..1
6. lim.'
,. ..
-tr.r.-
-
.r _
l
(A) -2
,.
,7
(
;u;.;;;:
-
(A) *oo
..
..
(D) 2
(c)
(E)
uonexisrent
is
(B)
_I
(c) o
(D)
3
(E)
6
-j.rr + 27
1t1ll
*,-"r
(A)3
9.
B)
'4 -1
5.r..t+ 27
,ttln
8.
IS
is
(B)@
(c)
1
(B) I
(c)
0
(D)
-1
(E) o
1-.t
llll z, i,
(A) *1
(D) 6
(E)
none of rhese
37
2E
38
Chapter 2: Limits and Continuity
10.
Iim
\'14
4
is
'tt
t
(A) -l
(D) F
(C) 0
(B) I
(E) none of these
11. If ["r] is the greatest integer not greater than x, then lim[-t] is
(A) =
2
(B) 1
1
L2.
(With the same notation)
(A) -3
13.
(B) -2
(C) nonexistent (D) 0
.-\rgtxl
(E)
none of these
is
(D) 0
(C) -1
(E)
none of these
The graph of y = arctan .r has
(A) vertical asymptotes at x = 0 and = n
(B) horizontal asymptotes aty = tt
(C) horizontal asymptotes aty -0 andy - ru
(D) vertical asymptotes at x = It
(E) none of these
-rr
14. limsinx
(A) is 1
(B) is infinity (C) oscillates between -l
(D) is zero (E) does not exist
15. Thegraphof
and
i
y-#nu,
(A) a vertical asymptote at x = 3 (B) a horizontal asymptote at.l' = J
(C) a removable discontinuity at x = 3 (D) an infinite discontinuity at x =
(E) none of these
16.
The funcrion,f(-r)
= {'"/''
l0
3
(r * 0)
(.r = 0)
(A) is continuous everywhere
(B) is continuous except at x = 0
(C) has a removable discontinuity at,r = 0
(D) has an infinite discontinuity at = 0
(E) has -r = 0 as a vertical asymptote
"r:
L7. Iim4T*r+0 -T'*J-{
(A)
I
is
)'
(B)
+
34
(c)
3
(D)
€
(E)
+
Set 2: Multiple-Choice Questions on Ltmtts ano uonununy
18.
.1
ft{_tstn
7 ts
(A) oo
19.
(B) i
(C) nonexistent (D)
Which statement is true about the curve
-1
(E)
none of these
.',
,
t-= -jjtiz+ir-4x?
(A) The line = -l is a vertical asyrnptote.
(B) The line -r" = I is a vertical asymptote.
(C) The line y = -+ is a horizontal asymptote.
(D) The graph has no vertical or horizontal asymptotes.
(E) The line I = 2 is a horizontal asymptote.
Questions 20 through 24 are based
on the function/shown in the graph
and defined below:
1-x (-t="<0)
?x'-2 (o <.r < l)
,,rr={ -x+2 (l<x<2)
(x =2)
1
2x-4 (2<r-<3)
tjg/(x)
(A)
(D)
(C)
(B) equals I
(E)
does not exist
none of these
equals
0
equals 2
21. The function/is defined on [-1,3]
(A)
(D)
'.,
if ;r'*
if .r *
(B)
(E)
0
3
if
.r
*1
(C)
it x *2
at each -r in [-1,3]
The function/has a removable discontinuity at
(A) x=0
(B) .r=l
(C) .r=2
(D) x=3
On which of the following intervals is/continuous?
(A) -1 <-r<0
(D) 2<x<3
(B) 0<x<1
(E)
noneofthese
(C) 1<;u<2
(E) none of these
40
Chapter 2: Limits and Continuity
?"4. The function/has
a
jump discontinuity at
(A) "r=-1 (B) x= I (C) x=2
(D) x=3 (E) none of these
2x2+l
z5' |96;6;;;
ts
(A) 4
26. rg3#
(B) nonexistent (C) 1
'hmxsinl
r+@
^
(A) 0
28.
(D) 2
(E)
nonexisrent
i'
(A) 0
n.
(C) 1
(B) -2
(D) -1
(E)
(B) oo (C) nonexistent (D) -1
(E)
none of rhese
is
I
g$i*
(A) 1
(B) 0
Ix2-l
29. Let f(x)={;;
(C) oo
(D) nonexistent (E)
none of rhese
irx*l
ifx=l
[+
Which of the following statements is (are) true?
L $/(x) exists. tr. /(1) exists. III. /is continuous at x=1.
(A) I only (B) II only (C) I and tr
(D) none of them (E) all of them
(
xz-x
30. r ]/ttl= zx for x#0,
*,
[,f(o) =
and if/is continuous at x = 0, then ft =
(B) -;
(A) -l
31.
(c) o
(D)
;
(E)
1
fr,',= #*rrorx+1,2,
I
Suppose {.f(r) =
I
[/(z)
-:,
= +'
Then/(x) is continuous
(A) exceptatx=1 (B) exceptatx=Z (C) exceptatx=1or2
(D) except at x= 0, 1, or 2 (E) at each real number
Set 2: Multiple-Choice Questions on Limits and Continuity
The graph
(A)
(B)
(C)
(D)
(E)
of /(x)
=
7f
nut
one vertical asymptote, at;= = 1
the y-axis as vertical asymptote
the x-axis as horizontal asymptote and x = +1 as vertical asymptotes
two vertical asymptotes, 3t r = +1, but no horizontal asymptote
no asymptote
Suppose
,{E /(.r)
f 6)
= -1,
following statements is
'1$
(are)
= -1, and f
(-3)
is not defined. Which of the
true?
I. f\ry,/(.r) = -t.
rse
il. /is continuous everywhere except at x = -3.
III. /has a removable discontinuity at x = -3.
(A)
(D)
(B) I only (C) III only
(E) All of thern
None of them
I and III only
Thegraphof
)vlr1wrl
y-#
tut
9Se
(A)
a
(B)
no horizontal asymptotes but two verlical asymptotes, at J = 0 and.r: = 1
ahorizontal asymptote atv= I andtwovertical asymptotes, atr=0andx=
(c)
(D)
horizontal asymptote at y
= "]
Uot no vertical asymptotes
horizontal asymptote at x = 2 but no vertical asymptotes
a horizontal asymptote at y = ] ana two vertical asymptotes, at x = +1
a
(E)
f -r: +.r
35. Letf(x)=l '
Ll
ifx+O.
if .r'= 0
Which of the following statements is (are) true?
I.
exists. II. {tg/t"t) exists. IIL f is continuous at;r = 0.
(A) I only (B) II only (C) I and II only
(D) all of them (E) none of then
/(0)
If y =
37.
I -, then
l*y it
2+10i
(A)0
(B)
lim
r+o
'j
(A)
i
(c)
*
LJ
(D)+
is
,T
-oo
(D) €
tg) ,b-L
v2\2
(E)
none of these
(c) -iz*L
(E)
nonexistent
1
42
Chapter 2: Limits and Continuity
Answers for Set 2: Limits
and Continuity
1.B
2. D
3.C
4. A
5.D
6. B
7. .A
8.E
17. B
18. C
19. A
20. A
2L. E
22. C
23. B
9. C
10.D
11.D
12.
E
13.8
L4. E
15. C
16.A
1. B.
2. D.
The limit as.r
3. C.
4. A.
Remove the common
5. D.
Note that
''*'
6,
7.
E.
g.
B.
A.
E.
C.
10. D.
11.
D.
Use the Rational Function Theorem (pages 30 and 31). The degrees of P(x)
and O(.x) are the same.
The fraction equals
i
factorx-
3 from numerator and denominator.
for all nonzero r.
8
(x-2)(.r-2+2x+4)
-;--;=
.u2_4 (.r:-2X-t+2)
.r3-
Use the Rational Function Theorem.
Use the Rational Function Theorem.
Use the Rational Function Theorem.
The fraction is equivalent
to
Since T) =2-'., therefore,
fr
;
"
; tfr" denominator approaches
t
---oo,
the fraction
-)
oo.
*m.
se.nigir.N2-1 on page33.
Note, from Figure N2-1, that
13. B.
L4. E.
Graph tan-r
r
,liq_[x] =
on your calculator in
As .r -> -, the function sin
does not exist.
r
[-5,
-3 but }q"[x]=
oscillates between
('i-lXtrj:) 'r+3
.-_l_
sinc, 3.r-9_
^"-? - 3(.r'_.3) -
16. A.
Notethat
t1
Nnre
(provided
-1
and 1, hence the lirnit
x*3),ycanbe definedto
I:-t =rifx*0andthat limf
-0.
t+0 "
fher :in't = lint;. ='--'.-1--r.1
-2.
5)x l-8,t1.
equal to 2 at x = 3, removing the discontinuity at that point.
E!
31. B
32. C
33. D
34. C
35. D
36. E
37. E
-) 2 is 0 + 8.
12. E.
ls. c.
24. B
25. B
26. B
27. E
28. A
29. C
30. B
.
be
Continuity
Answers for Set 2: Limits and
18. C.
As x -+
0,
{
43
takes on varying finite values as it increases. Since the sine
I
oscillates, taking on. infinitely many times, each
value between-1 and 1. The calculatorgraph of y, = sin (ii x) exhibits
this oscillating discontinuity at x = 0.
function repeats. sin
.B
j.c
19. A.
'.D
I.C
i.D
i.E
I,E
20.
A.
both x = 2 and .r: = -+
= O*#.,
asymptotes. Also. y = -+ is a horizontal asymptote.
Note that, since
-v
are veftical
ff"l = l1p "f(x) = 0.
[ry
21. E.
Verify that/is defined at x = 0,
[-1,3]).
22. C.
Note that
l,
2, and 3 (as well as at all other points in
[+ f "i = 1l]] "f (x) = 0. Howeve\ f (2) = 1. Redetining./(2) as 0
f
removes the discontinuity.
)f
P(r)
23. B.
24. B.
25. B.
26.
B.
The function is not continuous at
r
= 0, 1, or 2.
fgp/(x)=0*liq/(x):1.
2.rl+1 2x2+l
17;nZTt-- 4-F.
Use the Rational Function Theorem (pages 30 and 31).
Sincelxl=xifr>0butequals-xif.r.0, 11+*=
liml=
lim-'t--1.
"t
,r+0- ,t
:
Jl+i I
while
.r+0-
E.
28. A.
27.
29.
C.
30. B.
.1
Note that r sin { can be rewritten ur :'I
(n-;r) -+ 0.
Sincel(.x) =.r * 1 if r * 1,
\t3lftrl
r -) *" -} +
0.
As x -+ n,
-f(x)=+/=+.
must equal.no).
31. B.
and that, as
Only
_!g
-.r
exists (and is equal to 2).
forall-r*0. For./tobecontinuous atr=0, liqrl'(-r)
{ig f(r) = -l
.
= 1 and.r = 2 need be checked. Since .f(;r') = *
/t-r) = -3 = .f(i) ,l
is continuous at .r = 1. Since liry
for x
/(.r)
* I, 2. and
does not exist,
.f is not continuous at x = 2.
he
limit
32. C.
As x ->
,r
rbe
-)
-r1 ,
t-,.1="f(r) -)
0, so the -r-axis is a horizontal asymptote. Also,
.I j -, so ir = +1 are vertical asymptotes.
as
33. D.
No information is given about the domain of/except in the neighborhood
ofx = -3.
34. C.
As "r -+
-,.v J
]
at-tr=0andat.r=
35. D.
;
tn" denominator (but not the numelator) of y equals 0
1.
The function is defined at 0 to be 1, which is also
jl
tt:*
= Iim(.r+l).
Chapter 2: Limits and Continuity
36.
E.
Asr-+
l
0*, 10i-+ooandthereforey-+0.Asx-+
0, f +-oo,so
I
to:i-+ 0 and therefbre
equal. the
37.
E.
limit
-
+.Because
rhe two
on"-rio"lrimits
are nor
does not exist.
Asx-+ 0-,arctan 1 -+ -*,roy
).
L
- ^f,j.
y 2
The graph has a jump discontinuity at x =
Asx-+
0',)-).[2.
'l-" 2'
0. (verify with a calculator.)
A.
\
Bay Math League
Meet I
200s -2006
Round 4
Limits
4
Sbxt'
L
{*
sin 5x -)
..
llm_
r+a 5in
Jy
o
I
\,
Xtw
-
\d
lf*
3.
lim
v+4
w
64- yt
:
ru,t
i;,^^
YT\
1-C-G
)
t-3
f.ut {r
3 cus 3x
t
3
-L)(G y L)
-t2-r)(*"1*,rA
f,-z-
=
[,*
*1
7
"{
Answers:
(2 pts)
1.
(3 pts)
2.
(5 pts)
J.
cp
€7_
/.1
_ tqL
Number:
Name:
School:
SCORE:
-qy
-qr}(Vr+.{U+(r"
BAY IVIATH
LEAGUE
MEET
1
ROUND 4
Limits
Note: Leave all-answers in simplified radical form or as a fraction in
terms of zr(no decimals)!
r --L
1.,rarjlq4J;.'/v*G=ye.So x-a vn.+,[o -C<-":.]({-4T)= ,U*,fr-fr*E*
q!
3x
2.Findfi*I-?
;31_1=
x3
r-cos2x
3Find,.
----;==
x
ANSWERS:
X-3 fr _ _3x __-t(i; =-1
3-x
3
x
4 X = t+.sg-r'
i
it/ /
1.
/ ?'la-
(2
0ts)
(3
pts) 2.
-?
(5
pts)
t.
C
Number
Name
School
Score
i,o=o
MEET
BAY MATH LEAGTIE
1
ROLIND 4
Limits
Note: Leave all answers in simplified radical form or as a fraction in
terms of zr(no decimals)!
1. Find llp
2. Find liry1
/+ #
1+ sinx
- o
o)
---<-
?^
ii
L
X +J{-h4ft *X
h
(x+h)2 -x2
h--+0
3
+t
Find lim ^l;
t*.
,fy.{
zt
*'i
-{
i;_r
+V 2y
"__)_---_.
r.i
x'-x
x+l
x+t
,/
-J2.
.;
-
;l']
1€
. ','-i
;*l!*
,
v,+;
(3 pts)
;;
$
'-'"
!!
1.
,JN
'!
(5 pts)
r
J.
l"',.'l^-
1r+ I s-
ir
R)!
-!:-i;rr
":: )X
l_ -/
-
7/
i[;ti,rfi
ANSWERS:
(2 pts)
L
i
,'-:
rF
r
',
Y{tV+i I L,'.j
'r