AP Calculus AB Chapter 1 Limits - Mr. Morimotomrmori.weebly.com/uploads/3/7/6/6/37660747/ch_1_assignments.pdf · 1.1 Limits Numerical & Graphical Show all of your work on ANOTHER

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


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( )




(… )


limx:c

ƒsxd … limx:c

g sxd .


Exercises 2.2



a. b. c. d.


a. b. c. d.


a. exists.

b.

c.

d.

e.



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


a. does not exist.

b.

c. does not exist.




5. 6.




limx:0 ƒsxd?[-1, 1] .


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)



limt→−2

f t( ) limt→−1

f t( ) limt→0

f t( )

y = f x( )




(… )


limx:c

ƒsxd … limx:c

g sxd .


Exercises 2.2



a. b. c. d.


a. b. c. d.


a. exists.

b.

c.

d.

e.



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


a. does not exist.

b.

c. does not exist.




5. 6.




limx:0 ƒsxd?[-1, 1] .


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)



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




(… )


limx:c

ƒsxd … limx:c

g sxd .


Exercises 2.2



a. b. c. d.


a. b. c. d.


a. exists.

b.

c.

d.

e.



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


a. does not exist.

b.

c. does not exist.




5. 6.




limx:0 ƒsxd?[-1, 1] .


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)



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.


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.


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 .

AP Calculus AB Chapter 1 Limits - Mr. Morimotomrmori.weebly.com/uploads/3/7/6/6/37660747/ch_1_assignments.pdf · 1.1 Limits Numerical & Graphical Show all of your work on ANOTHER

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