APCalculusABChapter1Limits
SY:2016–2017Mr.Kunihiro
1.1LimitsNumerical&GraphicalShowallofyourworkonANOTHERSHEETofFOLDERPAPER.InExercises1and2,astoneistossedverticallyintotheairfromgroundlevelwithaninitialvelocityof15m/s.Itsheightattime is m.1.Computethestone’saveragevelocityoverthetimeinterval andindicatethecorrespondingsecantlineonasketchofthegraphof .2.Computethestone’saveragevelocityoverthetimeintervals , ,
and , , ,andthenestimatetheinstantaneousvelocityat .(Roundyouranswerstofourdecimalplaces.)InExercise3,usethefollowinggraphprovidedtohelpyouanswerthefollowingquestions.NOTE:PAYATTENTIONtotheVOCABULARY!3.ThefigurebelowshowstheestimatednumberNofinternetusersinChile,basedonthedatafromtheUnitedNationsStatisticsDivision.(a)EstimatetherateofchangeofNat .(b)Doestherateofchangeincreaseordecreaseas increases?Explaingraphically.(c)LetRbetheaveragerateofchangeover .ComputeR.(d)Istherateofchangeat greaterthanorlessthantheaveragerateR?Explaingraphically.
t h t( ) = 15t − 4.9t 2
0.5,2.5[ ]h t( )
1,1.01[ ] 1,1.001[ ]1,1.0001[ ] 0.99,1[ ] 0.999,1[ ] 0.9999,1[ ]
t = 1
t = 2003.5
t
2001,2005[ ]
t = 2002
InExercises4and5,findthefollowinglimits,orexplainwhytheydonotexist.4
. (a) (b) (c)
5
. (a) (b) (c)
InExercise6,usethegraphbelowtoanswerthefollowinglimitquestions.6.Whichofthefollowingstatementsaboutthefunction graphedherearetrue,andwhicharefalse?
(a) exists (b) (c)
(d) (e)
(f) existsateverypoint intheinterval
2.2 Limit of a Function and Limit Laws 73
The assertion resulting from replacing the less than or equal to inequality by thestrict less than inequality in Theorem 5 is false. Figure 2.14a shows that for
but in the limit as equality holds.u: 0,- ƒ u ƒ 6 sin u 6 ƒ u ƒ ,u Z 0,(6 )
(… )
THEOREM 5 If for all x in some open interval containing c, exceptpossibly at itself, and the limits of ƒ and g both exist as x approaches c,then
limx:c
ƒsxd … limx:c
g sxd .
x = cƒsxd … g sxd
Exercises 2.2
Limits from Graphs1. For the function g(x) graphed here, find the following limits or
explain why they do not exist.
a. b. c. d.
2. For the function ƒ(t) graphed here, find the following limits or ex-plain why they do not exist.
a. b. c. d.
3. Which of the following statements about the function graphed here are true, and which are false?
a. exists.
b.
c.
d.
e.
f. exists at every point in
g. does not exist.limx:1
ƒsxd
s -1, 1d .x0limx:x0
ƒsxdlimx:1
ƒsxd = 0
limx:1
ƒsxd = 1
limx:0
ƒsxd = 1
limx:0
ƒsxd = 0
limx:0
ƒsxd
y = ƒsxd
t
s
1
10
s ! f (t)
–1
–1–2
limt: -0.5
ƒstdlimt:0
ƒstdlimt: -1
ƒstdlimt: -2
ƒstd
3x
y
2
1
1
y ! g(x)
limx:2.5
g sxdlimx:3
g sxdlimx:2
g sxdlimx:1
g sxd
4. Which of the following statements about the function graphed here are true, and which are false?
a. does not exist.
b.
c. does not exist.
d. exists at every point in
e. exists at every point in (1, 3).
Existence of LimitsIn Exercises 5 and 6, explain why the limits do not exist.
5. 6.
7. Suppose that a function ƒ(x) is defined for all real values of x ex-cept Can anything be said about the existence of
Give reasons for your answer.
8. Suppose that a function ƒ(x) is defined for all x in Cananything be said about the existence of Give rea-sons for your answer.
limx:0 ƒsxd?[-1, 1] .
limx:x0 ƒsxd?x = x0 .
limx:1
1
x - 1limx:0
xƒ x ƒ
x
y
321–1
1
–1
–2
y ! f (x)
x0limx:x0
ƒsxd
s -1, 1d .x0limx:x0
ƒsxdlimx:1
ƒsxdlimx:2
ƒsxd = 2
limx:2
ƒsxd
y = ƒsxd
x
y
21–1
1
–1
y ! f (x)
Another important property of limits is given by the next theorem. A proof is given inthe next section.
7001_AWLThomas_ch02p058-121.qxd 10/1/09 2:33 PM Page 73
limx→1
g x( ) limx→2
g x( ) limx→3
g x( )
2.2 Limit of a Function and Limit Laws 73
The assertion resulting from replacing the less than or equal to inequality by thestrict less than inequality in Theorem 5 is false. Figure 2.14a shows that for
but in the limit as equality holds.u: 0,- ƒ u ƒ 6 sin u 6 ƒ u ƒ ,u Z 0,(6 )
(… )
THEOREM 5 If for all x in some open interval containing c, exceptpossibly at itself, and the limits of ƒ and g both exist as x approaches c,then
limx:c
ƒsxd … limx:c
g sxd .
x = cƒsxd … g sxd
Exercises 2.2
Limits from Graphs1. For the function g(x) graphed here, find the following limits or
explain why they do not exist.
a. b. c. d.
2. For the function ƒ(t) graphed here, find the following limits or ex-plain why they do not exist.
a. b. c. d.
3. Which of the following statements about the function graphed here are true, and which are false?
a. exists.
b.
c.
d.
e.
f. exists at every point in
g. does not exist.limx:1
ƒsxd
s -1, 1d .x0limx:x0
ƒsxdlimx:1
ƒsxd = 0
limx:1
ƒsxd = 1
limx:0
ƒsxd = 1
limx:0
ƒsxd = 0
limx:0
ƒsxd
y = ƒsxd
t
s
1
10
s ! f (t)
–1
–1–2
limt: -0.5
ƒstdlimt:0
ƒstdlimt: -1
ƒstdlimt: -2
ƒstd
3x
y
2
1
1
y ! g(x)
limx:2.5
g sxdlimx:3
g sxdlimx:2
g sxdlimx:1
g sxd
4. Which of the following statements about the function graphed here are true, and which are false?
a. does not exist.
b.
c. does not exist.
d. exists at every point in
e. exists at every point in (1, 3).
Existence of LimitsIn Exercises 5 and 6, explain why the limits do not exist.
5. 6.
7. Suppose that a function ƒ(x) is defined for all real values of x ex-cept Can anything be said about the existence of
Give reasons for your answer.
8. Suppose that a function ƒ(x) is defined for all x in Cananything be said about the existence of Give rea-sons for your answer.
limx:0 ƒsxd?[-1, 1] .
limx:x0 ƒsxd?x = x0 .
limx:1
1
x - 1limx:0
xƒ x ƒ
x
y
321–1
1
–1
–2
y ! f (x)
x0limx:x0
ƒsxd
s -1, 1d .x0limx:x0
ƒsxdlimx:1
ƒsxdlimx:2
ƒsxd = 2
limx:2
ƒsxd
y = ƒsxd
x
y
21–1
1
–1
y ! f (x)
Another important property of limits is given by the next theorem. A proof is given inthe next section.
7001_AWLThomas_ch02p058-121.qxd 10/1/09 2:33 PM Page 73
limt→−2
f t( ) limt→−1
f t( ) limt→0
f t( )
y = f x( )
2.2 Limit of a Function and Limit Laws 73
The assertion resulting from replacing the less than or equal to inequality by thestrict less than inequality in Theorem 5 is false. Figure 2.14a shows that for
but in the limit as equality holds.u: 0,- ƒ u ƒ 6 sin u 6 ƒ u ƒ ,u Z 0,(6 )
(… )
THEOREM 5 If for all x in some open interval containing c, exceptpossibly at itself, and the limits of ƒ and g both exist as x approaches c,then
limx:c
ƒsxd … limx:c
g sxd .
x = cƒsxd … g sxd
Exercises 2.2
Limits from Graphs1. For the function g(x) graphed here, find the following limits or
explain why they do not exist.
a. b. c. d.
2. For the function ƒ(t) graphed here, find the following limits or ex-plain why they do not exist.
a. b. c. d.
3. Which of the following statements about the function graphed here are true, and which are false?
a. exists.
b.
c.
d.
e.
f. exists at every point in
g. does not exist.limx:1
ƒsxd
s -1, 1d .x0limx:x0
ƒsxdlimx:1
ƒsxd = 0
limx:1
ƒsxd = 1
limx:0
ƒsxd = 1
limx:0
ƒsxd = 0
limx:0
ƒsxd
y = ƒsxd
t
s
1
10
s ! f (t)
–1
–1–2
limt: -0.5
ƒstdlimt:0
ƒstdlimt: -1
ƒstdlimt: -2
ƒstd
3x
y
2
1
1
y ! g(x)
limx:2.5
g sxdlimx:3
g sxdlimx:2
g sxdlimx:1
g sxd
4. Which of the following statements about the function graphed here are true, and which are false?
a. does not exist.
b.
c. does not exist.
d. exists at every point in
e. exists at every point in (1, 3).
Existence of LimitsIn Exercises 5 and 6, explain why the limits do not exist.
5. 6.
7. Suppose that a function ƒ(x) is defined for all real values of x ex-cept Can anything be said about the existence of
Give reasons for your answer.
8. Suppose that a function ƒ(x) is defined for all x in Cananything be said about the existence of Give rea-sons for your answer.
limx:0 ƒsxd?[-1, 1] .
limx:x0 ƒsxd?x = x0 .
limx:1
1
x - 1limx:0
xƒ x ƒ
x
y
321–1
1
–1
–2
y ! f (x)
x0limx:x0
ƒsxd
s -1, 1d .x0limx:x0
ƒsxdlimx:1
ƒsxdlimx:2
ƒsxd = 2
limx:2
ƒsxd
y = ƒsxd
x
y
21–1
1
–1
y ! f (x)
Another important property of limits is given by the next theorem. A proof is given inthe next section.
7001_AWLThomas_ch02p058-121.qxd 10/1/09 2:33 PM Page 73
limx→0
f x( ) limx→0
f x( ) = 0 limx→0
f x( ) = 1
limx→1
f x( ) = 1 limx→1
f x( ) = 0
limx→x0
f x( ) x0 −1,1( )
1.2One–SidedLimitsShowallofyourworkonANOTHERSHEETofFOLDERPAPER.InExercises1through4,usethegraphbelowtoanswerthefollowinglimitquestions.1.Whichofthefollowingstatementsaboutthefunction y = f x( ) graphedherearetrue,andwhicharefalse?
(a) lim
x→2f x( ) doesnotexist. (b) lim
x→2f x( ) = 2 (c) lim
x→1f x( ) doesnotexist
(d) lim
x→x0f x( ) existsateverypoint x0 intheinterval −1,1( ) .
(e) lim
x→x0f x( ) existsateverypoint x0 intheinterval 1,3( ) .
2.Whichofthefollowingstatementsaboutthefunction y = f x( ) graphedherearetrue,andwhicharefalse?
(a) lim
x→−1+f x( ) = 1 (b) lim
x→2f x( ) doesnotexist (c) lim
x→2f x( ) = 2
(d) lim
x→1−f x( ) = 2 (e) lim
x→1+f x( ) = 1 (f) lim
x→1f x( ) doesnotexist
(g) lim
x→0+f x( ) = lim
x→0−f x( ) (h) lim
x→cf x( ) existsatevery c intheopeninterval −1,1( )
(i) lim
x→cf x( ) existsatevery c intheopeninterval 1,3( ) (j) lim
x→−1−f x( ) = 0
(k) lim
x→3+f x( ) doesnotexist
2.2 Limit of a Function and Limit Laws 73
The assertion resulting from replacing the less than or equal to inequality by thestrict less than inequality in Theorem 5 is false. Figure 2.14a shows that for
but in the limit as equality holds.u: 0,- ƒ u ƒ 6 sin u 6 ƒ u ƒ ,u Z 0,(6 )
(… )
THEOREM 5 If for all x in some open interval containing c, exceptpossibly at itself, and the limits of ƒ and g both exist as x approaches c,then
limx:c
ƒsxd … limx:c
g sxd .
x = cƒsxd … g sxd
Exercises 2.2
Limits from Graphs1. For the function g(x) graphed here, find the following limits or
explain why they do not exist.
a. b. c. d.
2. For the function ƒ(t) graphed here, find the following limits or ex-plain why they do not exist.
a. b. c. d.
3. Which of the following statements about the function graphed here are true, and which are false?
a. exists.
b.
c.
d.
e.
f. exists at every point in
g. does not exist.limx:1
ƒsxd
s -1, 1d .x0limx:x0
ƒsxdlimx:1
ƒsxd = 0
limx:1
ƒsxd = 1
limx:0
ƒsxd = 1
limx:0
ƒsxd = 0
limx:0
ƒsxd
y = ƒsxd
t
s
1
10
s ! f (t)
–1
–1–2
limt: -0.5
ƒstdlimt:0
ƒstdlimt: -1
ƒstdlimt: -2
ƒstd
3x
y
2
1
1
y ! g(x)
limx:2.5
g sxdlimx:3
g sxdlimx:2
g sxdlimx:1
g sxd
4. Which of the following statements about the function graphed here are true, and which are false?
a. does not exist.
b.
c. does not exist.
d. exists at every point in
e. exists at every point in (1, 3).
Existence of LimitsIn Exercises 5 and 6, explain why the limits do not exist.
5. 6.
7. Suppose that a function ƒ(x) is defined for all real values of x ex-cept Can anything be said about the existence of
Give reasons for your answer.
8. Suppose that a function ƒ(x) is defined for all x in Cananything be said about the existence of Give rea-sons for your answer.
limx:0 ƒsxd?[-1, 1] .
limx:x0 ƒsxd?x = x0 .
limx:1
1
x - 1limx:0
xƒ x ƒ
x
y
321–1
1
–1
–2
y ! f (x)
x0limx:x0
ƒsxd
s -1, 1d .x0limx:x0
ƒsxdlimx:1
ƒsxdlimx:2
ƒsxd = 2
limx:2
ƒsxd
y = ƒsxd
x
y
21–1
1
–1
y ! f (x)
Another important property of limits is given by the next theorem. A proof is given inthe next section.
7001_AWLThomas_ch02p058-121.qxd 10/1/09 2:33 PM Page 73
90 Chapter 2: Limits and Continuity
Solution
(a) Using the half-angle formula we calculate
(b) Equation (1) does not apply to the original fraction. We need a 2x in the denominator,not a 5x. We produce it by multiplying numerator and denominator by :
EXAMPLE 6 Find .
Solution From the definition of tan t and sec 2t, we have
Eq. (1) and Example 11bin Section 2.2= 1
3 (1)(1)(1) = 13.
limt:0
tan t sec 2t
3t = 13 lim
t:0 sin t
t# 1cos t
# 1cos 2t
limt:0
tan t sec 2t
3t
= 25 s1d = 2
5
= 25 lim
x:0 sin 2x
2x
limx:0
sin 2x
5x = limx:0
s2>5d # sin 2x
s2>5d # 5x
2>5Eq. (1) and Example 11ain Section 2.2 = - s1ds0d = 0.
Let u = h>2. = - limu:0
sin uu
sin u
limh:0
cos h - 1
h= lim
h:0-
2 sin2 sh>2dh
cos h = 1 - 2 sin2sh>2d ,
Exercises 2.4
Finding Limits Graphically1. Which of the following statements about the function
graphed here are true, and which are false?
a. b.
c. d.
e. f.
g. h.
i. j.
k. l.
2. Which of the following statements about the function graphed here are true, and which are false?
y = ƒsxd
limx:2+
ƒsxd = 0limx: -1-
ƒsxd does not exist .
limx:2-
ƒsxd = 2limx:1
ƒsxd = 0
limx:1
ƒsxd = 1limx:0
ƒsxd = 1
limx:0
ƒsxd = 0limx:0
ƒsxd exists.
limx:0-
ƒsxd = limx:0+
ƒsxdlimx:0-
ƒsxd = 1
limx:0-
ƒsxd = 0limx: -1+
ƒsxd = 1
x
y
21–1
1
0
y ! f (x)
y = ƒsxd
a. b. does not exist.
c. d.
e. f. does not exist.
g.
h. exists at every c in the open interval
i. exists at every c in the open interval (1, 3).
j. k. does not exist.limx:3+
ƒsxdlimx: -1-
ƒsxd = 0
limx:c
ƒsxd
s -1, 1d .limx:c
ƒsxd
limx:0+
ƒsxd = limx:0-
ƒsxd
limx:1
ƒsxdlimx:1+
ƒsxd = 1
limx:1-
ƒsxd = 2limx:2
ƒsxd = 2
limx:2
ƒsxdlimx: -1+
ƒsxd = 1
x
y
0
1
2
1–1 2 3
y ! f (x)
Now, Eq. (1) applies withu = 2x.
7001_AWLThomas_ch02p058-121.qxd 10/1/09 2:34 PM Page 90
3.Let f x( ) =3− x, x < 2x2+1, x > 2
⎧⎨⎪
⎩⎪
(a)Find lim
x→2+f x( ) and lim
x→2−f x( ) .
(b)Does limx→2
f x( ) exist?Ifso,whatisit?Ifnot,whynot?(c)Find lim
x→4−f x( ) and lim
x→4+f x( )
(d)Does limx→4
f x( ) exist?Ifso,whatisit?Ifnot,whynot?
4.Let f x( ) =0, x ≤ 0
sin 1x
, x > 0
⎧⎨⎪
⎩⎪
(a)Does lim
x→0+f x( ) exist?Ifso,whatisit? Ifnot,whynot?
(b)Does limx→0−
f x( ) exist?Ifso,whatisit?Ifnot,whynot?(c)Does lim
x→0f x( ) exist?Ifso,whatisit?Ifnot,whynot?
5.(CalculatorUse)Let f x( ) = xx
(a)Graph f x( ) ontheinterval−3≤ x ≤ 3 .Is f x( ) undefinedwithinthisinterval?(b)Nowfind lim
x→0−f x( ) & lim
x→0+f x( ) . (c)Whatis lim
x→0f x( )
6.(MultipleChoice)
Thegraphofthefunction f isshowninthefigureabove.Whichofthefollowingstatementsabout f istrue?(A) lim
x→af x( ) = lim
x→bf x( ) (B) lim
x→af x( ) = 2 (C) lim
x→bf x( ) = 2
(D) lim
x→bf x( ) = 1 (E) lim
x→af x( ) doesnotexist
1.3FindingLimitsAnalytically(Part1)ShowallofyourworkonANOTHERSHEETofFOLDERPAPER.MultipleChoice
1. limx→2
x2 + x − 62 − x
is
(A)5 (B)3 (C)−3 (D)−5 (E)DNE
2. limx→9
x − 5 − 2x − 9
is
(A) 14 (B)− 1
4 (C)1 (D)0 (E)DNE
3. limx→2
1x− 12
x − 2is
(A) 14 (B)− 1
4 (C)1 (D)−1 (E)DNE
4. limx→1
tan−1 xsin−1 x +1
is
(A)0 (B) 14 (C) 1
2 (D) π
2 (E) π
2π + 4
Forproblems5&6,usethetableprovidedbelow.5.Giventhefollowingselectedvaluesforcontinuousfunctions f x( ) and g x( ) inthetablebelow:
limx→3
f g x( )( )g f x( )( ) is
(A) 14 (B) 1
3 (C)1 (D)3 (E)4
x 1 2 3 4f x( ) 4 2 3 1g x( ) 2 3 1 4
6. limx→2
f g−1 x( )( )g f −1 x( )( ) is
(A) 43 (B)1 (C) 3
4 (D)3 (E)4
FreeResponseFindthelimitsforeachofthefollowing.
7. limx→0
7 + sec2 x 8. limx→0
1+ x + sin x3cos x
9. limx→1
x4 −1x3 −1
10. limx→−2
x + 2x2 + 5 − 3
11. limx→−3
2 − x2 − 5x + 3
12. limx→4
4x − x2
2 − x
13.Suppose lim
x→bf x( ) = 7 and lim
x→bg x( ) = −3 .Find
(a) limx→b
f x( ) + g x( )( ) (b) limx→b
f x( ) ⋅g x( ) (c) lim
x→b4g x( ) (d) lim
x→bf x( ) g x( )
14.Thegraphsof f x( ) = x , g x( ) = −x ,andh x( ) = xcos 50πx
⎛⎝⎜
⎞⎠⎟ on
theinterval−1≤ x ≤1 aregivenattheright.UsetheSqueeze
Theoremtofind limx→0
xcos 50πx
⎛⎝⎜
⎞⎠⎟ .Justifyyouranswer.
15.If1≤ f x( ) ≤ x2 + 2x + 2 forall x ,find lim
x→−1f x( ) .Justifyyouranswer.
16.If−3cos π x( ) ≤ f x( ) ≤ x3 + 2 ,evaluate lim
x→1f x( ) .Justifyyouranswer.
1.4FindingLimitsAnalytically(Part2)ShowallofyourworkonANOTHERSHEETofFOLDERPAPER.MultipleChoice
1. limx→0
sin2xxcos x
is
(A)0 (B)1 (C) 12 (D)2 (E)DNE
2. limx→0
cos2 x −12xsin x
is
(A)−1 (B)− 12 (C)1 (D) 1
2 (E)0
3. limx→0
cot 6xcsc3x
is
(A)2 (B)0 (C) 12 (D)−2 (E)DNE
4. limx→0
sinα cos x −1( )− cosα sin xx
is
(A)1 (B) cosα (C) sinα (D)−sinα (E)DNE
5. limx→0
x ex + 1x
⎛⎝⎜
⎞⎠⎟ is
(A)0 (B)1 (C)2 (D)DNE (E)Noneoftheabove
6. limx→−3+
x2 x + 3x2 − 9
is
(A)0 (B)1 (C)−1.5 (D)1.5 (E)DNE
7. limx→0
tan3x2x
is
(A)0 (B) 12 (C) 2
3 (D) 3
2 (E)Noneoftheabove
FreeResponseEvaluatethefollowinglimits.Note:thesymbol x⎢⎣ ⎥⎦ isthegreatestintegerfunction.
8. limx→−2−
x + 3( ) x + 2x + 2
9. limx→1+
2x x − 2( )x − 3
10. limx→3+
x⎢⎣ ⎥⎦x
11. limx→4−
x − x⎢⎣ ⎥⎦( ) 12. limx→06x2 cot x( ) csc2x( ) 13. lim
x→0
tan xcos x2x
14. limx→0
tan3xcsc8x( ) 15. limx→0
sin x5x2 − x
16. limx→0
sin3 xx3 1+ cos x( )
17.Explainwhy limx→0
xxdoesnotexist.
1.5LimitsInvolvingInfinityShowallofyourworkonANOTHERSHEETofFOLDERPAPER.MultipleChoice
1.Thegraphof y = x2 − 93x − 9
has
(A)averticalasymptoteat x = 3 (B)ahorizontalasymptoteat y = 13
(C)aremovablediscontinuityat x = 3 (D)aninfinitediscontinuityat x = 3 (E)Noneoftheabove
2.Whichstatementistrueaboutthecurve y = 2x2 + 42 + 7x − 4x2
?
(A)Theline x = − 14isaverticalasymptote.
(B)Theline x = 1 isaverticalasymptote.
(C)Theline y = − 14isahorizontalasymptote.
(D)Thegraphhasnoverticalorhorizontalasymptote.(E)Theline y = 2 isahorizontalasymptote.
3. limx→∞
2x2 +12 − x( ) 2 + x( ) is
(A)−4 (B)−2 (C)1 (D)2 (E)DNE
4. limx→∞
2− x
2xis
(A)−1 (B)1 (C)0 (D)∞ (E)DNE
5. limx→−∞
2− x
2xis
(A)−1 (B)1 (C)0 (D)∞ (E)DNE
6. limx→−∞
5x3 + 2720x2 +10x + 9
is
(A)−∞ (B)−1 (C)0 (D)3 (E)∞
FreeResponseEvaluatethefollowinglimits.
7. limx→∞
2x3 + 7x3 − x2 + x + 7
8. limx→−∞
9x4 + x2x4 + 5x2 − x + 6
9. limx→∞
10x5 + x4 + 31x6
10. limx→∞
8x2 − 32x2 + x
11. limx→−∞
x2 − 5xx3 + x − 2
12. limx→−∞
x−1 + x−4
x−2 − x−3
13. limx→−5−
3x2x + 5
14. limx→ −π 2( )−
sec x 15. limx→∞arctan x
16. limx→0−
1+ csc x( ) 17. limx→7
4x − 7( )2
18. limx→5
1x − 5
Sketchthegraphofafunction y = f x( ) thatsatisfiesthegivenconditions.Noformulasarerequired–justlabelthecoordinateaxesandsketchanappropriategraph.(Theanswersarenotunique,sotherearemultiplesolutions)19. f 0( ) = 0 , lim
x→±∞f x( ) = 0 , lim
x→0+f x( ) = 2 ,and lim
x→0−f x( ) = −2
20. f 2( ) = 1 , f −1( ) = 0 , lim
x→∞f x( ) = 0 , lim
x→0+f x( ) = ∞ , lim
x→0−f x( ) = −∞ ,and lim
x→−∞f x( ) = 1
1.6Continutiy&IVTShowallofyourworkonANOTHERSHEETofFOLDERPAPER.MultipleChoice1.Let g x( ) beacontinuousfunction.Selectedvaluesof g aregiveninthetablebelow.
Whatisthefewestnumberoftimes g x( ) willintersect y = 1 ontheclosedinterval3,10[ ]?(A)None (B)One (C)Two (D)Three (E)Four2.Leth x( ) beacontinuousfunction.Selectedvaluesof h aregiveninthetablebelow.
Forwhichvalueof k willtheequationh x( ) = 2
3haveatleasttwosolutionsonthe
closedinterval 2,7[ ]?
(A)1 (B) 34 (C) 7
9 (D) 2
3 (E) 11
18
3.If f x( ) =x +1, x ≤13+ ax2, x >1
⎧⎨⎩
,then f x( ) iscontinuousforall x ifa =?
(A)1 (B)−1 (C) 12 (D)0 (E)−2
4.If f x( ) =2x + 5 − x + 7
x − 2, x ≠ 2
k, x = 2
⎧⎨⎪
⎩⎪,andif f iscontinuousat x = 2 ,then k = ?
(A)0 (B) 16 (C) 1
3 (D)1 (E) 7
5
5.Let f bethefunctiondefinedbythefollowing:
f x( ) =
sin x, x < 0x2, 0 ≤ x <12 − x, 1≤ x < 2x − 3, x ≥ 2
⎧
⎨⎪⎪
⎩⎪⎪
Forwhatvaluesof x is f NOTcontinuous?(A)0only (B)1only (C)2only (D)0and2only (E)0,1,and26.Let f beacontinuousfunctionontheclosedinterval −3,6[ ] .If f −3( ) = −1 andf 6( ) = 3 ,thentheIntermediateValueTheoremguaranteesthat(A) f 0( ) = 0 (B)Theslopeofthegraphof f is 4
9somewherebetween−3 and6
(C)−1≤ f x( ) ≤ 3 forall x between−3 and6(D) f c( ) = 1 foratleastone c between−3 and6(E) f c( ) = 0 foratleastone c between−1and3
7.Let f bethefunctiongivenby f x( ) = x −1( ) x2 − 4( )x2 − a
.Forwhatpositivevaluesof
a is f continuousforallrealnumbers x ?(A)None (B)1only (C)2only (D)4only (E)1and4only8.If f iscontinuouson −4,4[ ] suchthat f −4( ) = 11 and f 4( ) = −11 ,thenwhichmustbetrue?(A) f 0( ) = 0 (B) lim
x→2f x( ) = 8
(C)Thereisatleastone c∈ −4,4[ ] suchthat f c( ) = 8 (D) lim
x→3f x( ) = lim
x→−3f x( )
(E)Itispossiblethat f isnotdefinedat x = 0 FreeResponse9.Atoycartravelsonastraightpath.Duringthetimeinterval 0 ≤ t ≤ 60 seconds,thetoycar’svelocity v ,measuredinfeetpersecond,isacontinuousfunction.Selectedvaluesaregivenbelow:
For0 < t < 60 ,musttherebeatimewhen v t( ) = −2 ?Justify.
10.Forthefunction f x( ) = x − 2( )2 , x = 45, 4 < x ≤10
⎧⎨⎪
⎩⎪.Find f 4( ) and f 10( ) .Doesthe
IVTguaranteea y -valueu on 4 ≤ x ≤10 suchthat f 4( ) < u < f 10( )?Whyorwhynot?Sketchthegraphof f x( ) foraddedvisualproof.11.Thefunctions f and g arecontinuousforallrealnumbers.Thetablebelowgivesvaluesofthefunctionsatselectedvaluesof x .Thefunction h isgivenbyh x( ) = g f x( )( ) + 2 .
Explainwhytheremustbeavaluew for1< w < 5 suchthat h w( ) = 0 .12.Thefunctions f and g arecontinuousforallrealnumbers.Thefunctionh isgivenby h x( ) = f g x( )( )− x .Thetablebelowgivesvaluesofthefunctionsatselectedvaluesof x .Explainwhytheremustbeavalueu for1< u < 4 suchthath u( ) = −1 .