Unidad 8 - Funciones, Límites y Continuidad - Problemas resueltos

8/11/2019 Unidad 8 - Funciones, Lmites y Continuidad - Problemas resueltos

1/21

1 BACHILLERATO - Matemticas - Unidad 8 - Funciones, !mites " continuidad

#$%18& 'I

#$%18& ' II a( =+ 03x ( ) NoExistem == 90=)( = 22xy 2=m 32 = xy '2663c( = 25y 0=m = 0..tgarc 0= d( ( ) += 3132 xy 32=m ( )= 32..tgarc '18146 e( = 02x ( ) NoExistem == 90=*( = 4xy 1=m = 1..tgarc 45=

#$%18& ' III a( ( )( ) 081 + xx 1 8 +1+x + +8x +

( )( )81 + xx + + [ ]8181 xRx )( ( ) 03 22 >+xx +

x + ( ) 22 3+x + + ( ) 22 3+xx + ( )+> 00xRx

#$%18 ' 1 a( ( ) = drticaFuncinCuaxf ( ) RfD =)( ( ) = rsaalidadInveproporcionxf ( ) [ ]2= RfD

#$%18 ' 2 a( ( ) [ )+= 0fD )( ( )( ) + 0111 2 xxx 11: xD

( )( ) + 01112 xxx 11: xxD ( ) [ ]11= RxfDc( ( ) RfD =

#$%18 ' 3 #o. e/em0o 10 x ctaRe

75 x SimpleCurva

#$%18 ' 4 a( ( ) ( )4=fD ( ) ( )3=fR)( ( ) [ ] [ ]5.25.110 =fD ( ) ( )11=fR

#$%182 ' 5 1341.650 13541.860 13642.300 1343.040 13844.150

#$%182 ' 6 La grfica c)% La 7ue incu"e un dec.ecimiento de a distancia, asta ce.o#o. tanto, es a 9nica 7ue .e0.esenta una :;ueta a casa


2/21


La tablaes e>tensa% La frmulaes en$o..osa%

La grficaes a ms con;eniente%

#$%182 ' 8 a( En ( ) ( )1=fD , e ( ) parbolafR =En ( ) ( )= 1fD , e ( ) rectafR =

)(

#a.a 1


3/21


x 1 5 1, 1,2 1, 1,1 1,@ 1,6 1,@& 1,@P ' + ' + ' + ' + ' +

( ) [ ]6036,16035,1101 f

#$%131 ' 13 a( x ',3 '1,1 ',33 '1,1 ',333 '1,1f ',&5@ ',625 ',&5& ',632& ',&@ ',6338 21. =lm

)(

x ,3 2%1 ,33 2,1 ,333 2,1 f 1,33@2 5,5 1,3336 5, 1,3333 5,1 2. =lm c( x @,3 6,1 @,33 6,1 @,333 6,1 f @,3268 6,568 @,332& 6,5& @,3332 6,5 4. =lm d( x ',1 ,1 ',1 ,1 ',1 ,1 f @,3@6 @,3@6 @,3 @,3 @, @, 3. =lm e( x 1,3 5,1 1,33 5,1 1,333 5,1

f 1,583 1,&&56 1,6 1,658@ 1,6158 1,61& 2. =lm*( x 2,3 8,1 2,33 8,1 2,333 8,1 f 11,3633 15,633 11,33& 15,& 11,333& 15,& 12. =lm

#$%131 ' 14 a( ( )=af 2 ( )=bf 0 ( )=cf NoExiste )( ( )= axlm 2 ( )= +axlm 2 ( )= axlm 2 ( )= bxlm 3 ( )= +bxlm 0 ( )= bxlm NoExiste ( )= cxlm + ( )= +cxlm NoExiste ( )= cxlm NoExiste

#$%135 ' 15 a( +

+=

+

+

222

2

2

11

11

1

1

x

xx

xx

x

xx

x

=++

1

0

01

00 ( )=+xlm 0

)(

+=

+

x

x

xx

x

xx

x

23

15

23

15

=+

3

5

03

05 ( )=+xlm

3

5

c( +

=

+ 17

1

7

1

7

7

7

1

7

1

!

!

x

x

x

x

==+

01

0

10

0 ( )=+xlm 0

d( +

+

=+

+

3

3

33

3

33

2

3

3

82

133

82

133

x

xx

xx

xxx

x

x

x

=+ + 23

02003 ( )=+xlm 2

3

e( +++

++

=

+++

++

432

43

444

2

4

3

444

4

1111

111

1

1

xxxx

xx

xx

x

x

x

x

x

xx

x

x

x

+==+++++

0

1

0000

001 =lm +

@=51


4/21


*( +

+=

+

+

32

32

333

3

33

326

13

326

13

xx

xx

xx

x

x

x

xx

x

==+

+0

6

0

006

00 ( )=+xlm 0

#$%135 ' 16 a( =xxx

xx

x

71

2

7

2

==+ 212

01

2

( )=xlm 2

)(

+=

+

7

72

77

7

77

5

13

12

13

12

x

xx

xx

x

xx

x

==+

03

0

03

00 ( )=xlm 0

c( ( )

( ) == xxxx

xx

3232

1

32

33 ( ) < + 01 ( )=xlm 0

d( +=

+

3

2

33

2

33

3

17

2

1

17

2

xx

x

xx

x

x

x

x

x

=

00

01 ( )=xlm

e( ( ) ( )

+x

x

xlmxlm72

1

2

7 0

1

+ ( )=xlm 0

*( ( ) ( )[ ]+ x

xxlmxlm 43

4

3 22

( )=xlm +

#$%13@ ' 17 a( ( ) =

0

12xlm )( ( ) = +

0

97xlm +

c( ( ) = +0

11xlm d( ( ) = +

++

0

33xlm +

e( ( ) = +

+

0

65xlm *( ( ) = +

0

44xlm +

#$%13@ ' 18 a( ( ) axlm + ( ) +axlm +)( ( ) axlm + ( ) +axlm NoExistec( ( ) axlm NoExiste ( ) +axlm +

#$%13& ' 19 a( =+

4

0

22

220 )(

( ) ( )

( ) ( ) +

=+

2

5

22

52

x

x

xx

xx

=+

4

3

22

52

4

3

#$%13& ' 20 a( ( ) ( )= glmflm =03 0 )( ( )

( ) ==0

3

glm

flm ( )0. =gSi"m

c( ( )[ ] ( ) == 03glmflm 1 d( ( )[ ] ( ) == 30flmglm 0

#$%13& ' 21

+

3

3

3

2

3

333

3

22

3

x

x

x

x

x

xx

x

x

x

=+

200

001

2

1

+

xx

x

xx

x

6

222

==+

1

0

01

000

6=51


5/21


#$%132 ' 22 a( #ER$IC%"ES 092 =x 3=x 3=x

( ) = +

0

73xlm ( ) =

+

0

73xlm +

( ) =

0

53xlm ( ) = +

+

0

53xlm +

ES&'RI('N$%"

22

2

22

9

12

xx

xxx

x

( ) 010x 0 0=y

')"IC*%S ( ) 0:9

122

=

xx

xxlm No$iene

)( #ER$IC%"ES =++ 0232 xx ( )( ) =++ 012 xx 2=x 1=x

( ) ( )( )

=

10

132xlm ( ) ( )( )

=

++

10

132xlm +

( ) ( )( )=

01

21xlm ( )

( )( )=

++

01

21xlm +

ES&'RI('N$%" ( )xlm ( ) ++++

001

003 No$iene

')"IC*%S ( )xlm ++

++

23

5273

2 xx

xx 73 = xy

c( #ER$IC%"ES ( )

0

50xlm 0=x

( ) =

0

50xlm ( ) =

+

+

0

50xlm +

ES&'RI('N$%" ( )xlm =+3

02

3

2

3

2=y

')"IC*%S ( ) 0:3

52=

+ x

x

xxlm No$iene

d( #ER$IC%"ES ( ) 0xlm +

0

10 0=x

ES&'RI('N$%" ( )=xlm No$iene

')"IC*%S ( ) +x

xxlm 1

4 xy 4=

#$%132 ' 23 a( #ER$IC%"ES ( ) =xlm No$iene

ES&'RI('N$%" ( )

=xlm

No$iene ')"IC*%S ( ) =+=

x

xxlmm

32

1

( )[ ]= mxxflmn + xx 32 =+

++

xx

xxxx

3

33

2

22 ( ) ( )

+

+

xx

xx

3

3

2

22

0

+= nmxy xy = )( #ER$IC%"ES ( ) 3xlm + 3=x

ES&'RI('N$%" ( ) +=xlm No$iene

')"IC*%S ( ) =+

=

+

=23

33

3

1:

3

1

xx

xlmx

x

xxlmm 1

&=51


6/21


( )( )

=

+

+

=

+

+

=

+

=

xx

xx

xlm

xx

x

xx

x

lmmxx

xxlmn

3

13

13

3

1

3

1

3

1

3

2

3

23

3

2

3

x 23+= xy +x 23+= xy

c( #ER$IC%"ES #a.a yRx 022

+x

No$ieneES&'RI('N$%" ( )=xlm 3 3=y

')"IC*%S ( ) 0:2

132

2

=

+

xx

xxlm No$iene

d( #ER$IC%"ES ( ) +=+ 013xlm =+ 013x 1=x ES&'RI('N$%" ( ) 0=xlm 0=y

')"IC*%S ( ) 0:13

=

+ x

x

xxlm No$iene

#$%133 ' 24 a(

polinmicaf

= continuaf

Rx

)( =1x f = continuaf ( )1 Rx c( = 012x = 1x f = continuaf ( )11= Rx d( =0x f = continuaf ( )0Rx e( Rx 012 +x = continuaf Rx=

*( ( )( )

43

2

xx

xf = 43x =f = continuaf ( )43= Rx

$( aParteEnterf EnteroxNoDefinida = =continuaf (Rx = ( polinmicaf =continuaf Rx , si ( ) ( )+ 11 xlmxlm

[ ] [ ]= xlmfxlm 2)1( 2 ( )[ ] [ ]=+ + 11 xlmfxlm 2

( )1

f 0

( ) ( ) ( ) idaddiscontinufxlmxlm = + 111 ( )1 Rx

#$%133 ' 25 a( ( )( )

+1

11

x

xxlm ( ) 21 =+xlm ( )=1f 2

)( ( ) ( )( ) ( )

++

22

21

xx

xxlm

4

3

2

1=

++

x

xlm ( )=2f

4

3

#$%133 ' 26 ( )( )

( )( )

+21

11

xx

xxlm =

+

=+

21

11

2

1

x

xlm 2 12 + +

2

3=+

#$%51 ' 27 a( ( ) ( )( )22

3253

nnlmnlm

+=+

2

3

)( ( )nnn

nlmnlm

2

21

+

+=+ =+ 01

11

c( ( ) 2222

1

11

nnn

nnlmnlm

++

=+ =++10

000

d( ( )nnnnn

nnnnnlmnlm

1

1

++

+=+ ==+

2

0

11

110

#$%51 ' 28 a( ( ) ( )


7/21


c( ( ) =+

+=+

nnn

nnnlmnlm

31

1

+

+611

11

nn

nlm =

++

00

01 + nteNoConverge

#$%51 ' 29 a( =na ( ) nn 1 =na ( ) ( )nnne 1)( =na n =na n c( =na n =na ( ) en 32

d( =na ( ) ( )71 n

n

=na ( )1

23

++ nnn

#$%51 ' 30 a( ( )

( ) =

++

++++

3

32

21

312

n

n

n

n

( )( )23

3

++ nn 0> 1+< nn aa Creciente

)( ( ) ++

=+n

nlmnlm

21

32=

1

22 2 Acotada superiormente

#$%5@ ' 31 a( ( ( ++

=++

+++

nnnn

nnnn

1

1

1

11

1

0

)( =++

+++nnnn

nnnnnnnn22

2222

3

33

nnnn

n

++

223

4

=++

++

=

2

4

0101

4

3

4

22

2

22

2

n

n

n

n

n

n

n

n 2

c( =++

+=

++

+++

nnn

n

nnn

nnnnnn

51

51

51

5151

2222

2222

++

+=

nn

n

5111

15

2

+=

+++

11

5

0101

05

2

5

#$%5@ ' 32 a( ( )

( )( )

+

+=

++

++

++ 152

215

21

11

1

21

n

nnn

nn

2e

)(

++=

++

+

+++ nn

n

nnn

nnnn

3

1

3

2

1

2

22

3

11

3

11 =0e 1

c( ( )

+

+=

++

+

+ 7

6

6

7

67

11

7

61

n

nn

n

nn

6e

d(

+=

+

32

32

3

11

3

11

n

n

nn

nn

2e

e(

++=

++

++

++ 72

72

7

11

7

11

n

nnn

nn=1e e

*( ( ) ( )

++

+=

+++

++

++

++ 1

5

5

3

2

1

3

2

22

53

11

3

51

n

n

n

n

n

n

nnn

n 1e

#$%5 ' 33 a( 012 +x ( ) =fD R)( = 01x 1=x ( )=fD { }1Rc( =+ 02x 2=x 042 x = 00f 2xf 4=f

En 0.inci0io, ( )=fD { }2R #e.o es admisi)e, =D R d( = 042x 2=x 2=x ( ) = 002f ( ) 21 xf 41=f

2=51


8/21


9/21


10/21


$(

+

2

2

34

12

x

x ==+

4

2

04

02

2

1(

+

3

32

11

15

x

xx ==+

1

0

01

000

#$%52 ' 46 a( +

+

2

214

51

xx

x

=++

004

01

2

1

)( +

x

xx

12

3383

3

==+

2

2

02

0083

1

c(

+

+=

++

x

x

x

xx

51

3

5

3

=+01 + d(

510

5

10

13

2

61

x

x

x

x

x ==

1010

#$%52 ' 47 a( = 00 0

)( ( )( )

( )

++

=

++

+++

xxxx

xxxx

3

3

3

33 =

30

c( ( ) +

=+

x

xx

x

11

1

1

12

==+ 1

1

01

11

d( ( )( )

++=

+++++

2

11

1111

11 xx

xxxx

xx =

+2

+

#$%52 ' 48 a( ( )=+xlm + ( )=xlm )( ( )=+xlm ( )=xlm +

#$%52 ' 49 a( ( )

2

1

3

x

x =0

2 )( ( ) =

xxx

x 1

1

1 =0

1

c( ( ) =

xxx

x 1

1

1 =+0

1 + d(

( )

=

1

1

1

12 xx

x=+

0

1 +

e( NoExiste ( )"""" gyf$ras *( ( )

2

33

xx

x=

+025

+

$( ( )

1

33

xx

x = 025 (

x

xx1

1

3

1 2 ( ) =1

1

#$%52 ' 50 a( ==+

2

10

24

645 )( ==

+6

18

39

993

c( =

=+

2

2

11

311 d(

( )

( ) =

+

+

=+

10

30

1

3

1

3

x

x

xx

xx3

e == 16952 4 *(( )( ) +=

+1

1

11x

x

xx2

$( == 0399 22x 0 ( ( ) =

22 399 x +0

i( ( )( )

( )( ) +

=+

2

1

21

11

x

x

xx

xx=

+21

112 /( ( )( )

+

=+

1

1

11

1

xxx

x

2

1

1=51


11/21


( ( )( )( )

( )( ) ( )( )++=

+

++310

33

3109xx

xx

xxx( ) ==++ 61939109 114

(( ) ( )

( )

+=

+

x

x

xx

xx 1

2

21==

+1

2

1

112

#$%52 ' 51 ( ) ( ) ( ) ( ) +=+=++ 323233 22

-x

-

-x-

-

xx-x-x32 x

#$%52 ' 52 ( )( ) ( )( )

( )( )

( )( )

+++

=+++

=++

+=

++

2222

21

2

11

12

11

2

11

3

1

1

xx

x

xxx

xx

xxx

xx

xxxx 1

#$%52 ' 53 a( 2xx

= 200

0 )( NoExiste ( )"""" dyc$ras

c(

21

1 =

= 1

1

21

11 d( ( ) + 112 ==+ 1212 1

e( ( ) = 16132 5 *( ++=++ ++++ 142144 7 $( ( ) = 18142 7 ( ++=++ 142144 7

#$%58 ' 54 a( Funcin 0oinmica No$iene )( #ER$IC%"ES y + 012x No$iene

ES&'RI('N$%" ( ) =+

+ 2

01

2

11

22

xxlm 2=y

')"IC*%S Gi 0m ( ) 0:1

22

2

+

= xx

xxlmm No$iene

c( #ER$IC%"ES y =+ 05x 5=xES&'RI('N$%" ( ) 0xlm 0=y

')"IC*%S Gi 0m ( )

+= 0:

5

3x

xxlmm No$iene

d( ( )( )

( )( ) +

=+

3

1

31

11

x

x

xx

xx #a.a 1x Funcin ! "efi!i"aen 1=x

#ER$IC%"ES y = 03x 3=x

ES&'RI('N$%" ( ) +

= 1341

112

2

xx

xxlm 1=y

')"IC*%S Gi 0m ( )

+

= 0:

34

12

2

xxx

xxlmm No$iene

e( #ER$IC%"ES y =+ 01x 1=xES&'RI('N$%" ( ) xlm No$iene

')"IC*%S Gi 0m ( )

+

= xx

xxlmm :

1

2

1

( ) ( ) +

=+

=

+=

xx

xx

x

xxlmn

11

1

11

1

2

1 1= xy

*(

#ER$IC%"ES

y

=++ 012

xx = imaginariox

No$iene

ES&'RI('N$%" x y No$iene

')"IC*%S Gi 0m ( )

++++

= xxx

xxxlmm :

1

722

3

1

11=51


12/21


( ) ( ) +++

=

++++

=1

731

1

722

2

2

3

xx

xxx

xx

xxxlmn 1 1= xy

$( #ER$IC%"ES y +0x 0=x JGo ado +0 (ES&'RI('N$%" ( ) xlm No$iene

')"IC*%S Gi 0m ( )

+= x

x

xxlmm :

13

+

1

11 3

x1

( ) ( ) =

+=

xx

x

xxlmn

11

13

0 xy=

( #ER$IC%"ES y 4x

+4

17112 2

x

xx 4=xES&'RI('N$%" ( ) xlm No$iene

')"IC*%S Gi 0m ( )

+= x

x

xxxlmm :

4

17112 2

2

( ) ( ) +

+

=4

1732

4

17112 2

x

xx

x

xxxlmn 3 32 = xy

#$%58 ' 55 ++=

cx

bay x ay 2y 2=a

y cx =1x 1=c

( )

+=16

263b

f

=16

23b

5=b

#$%58 ' 56

( )( )( )( )xx

xx

+

+31

2121

#ER$IC%"ES y 1=x 3=xES&'RI('N$%" x ylm. 2 2=y

')"IC*%S Gi 0m

( )

+

= 0:

23

212

2

xxx

xxlmm No$iene

#$%58 ' 57 ( )( )

( )( ) =

+

=51

52

xx

xxy

1

2

+

x

x ( ) ==

+

=6

3

15

255xlm

2

1 )( o#alorFinit

En consecuencia, 5=x !0uede se. as!ntota

#$%58 ' 58 ( )( )( )

( )( ) =

++

=11

311

xx

xxxy 3x

#a.a 1=x a" 7ue asi$na. e ;ao. ( ) = 311xlm 2#a.a 1=x a" 7ue asi$na. e ;ao. ( ) = 311xlm 4

#$%58 ' 59 a( ( ) polinmicaxf = dContinuida K Rx= )( 042 >+x dContinuida K Rx= c( = 02x 2=x dContinuida K { }2= Rx

d(

=+ 0522

xx 61=x dContinuida K { }6161 +Rx e( =++ 0452 xx 1=x 4=x dContinuida K { }14 Rx *( Rx 052 ++ xx dContinuida K Rx= $( 049 2x dContinuida K ( ] [ )+ 3232x

15=51


13/21


( + 0832 2 xx =4

733x

+

+

4

733

4

733x

#$%58 ' 60 a( A 0.io.i, ( )xf es continua, e>ce0to 0a.a 1=x , 0o. se. 0oinmica% ( ) 1xlm ( ) +=+ 861262x ( ) +++ =+=+ 87171 xxlm Lue$o, es tam)iDn continua en 1=x ( ) == ContinuaEnxf R

)( ( ) Continuapolinmicaxf = udas en e ;ao. 1=x L!mites ate.aes ( ) ( ) += 86121xlm ( ) ( ) ++ = 6711xlm Gon distintos #o. tanto, ( )xf no es continua en 1=x =dEnContinuida { }1R c( iscontinuidad de sato in*inito en 0=x L!mites ate.aes en 1=x

( ) ( )

( ) =

+

1

6121xlm 8 ( ) ( ) =+ ++ 71lxlm 8 Continuidad en 1=x

En consecuencia, =dEnContinuida { }0R

d(

i$e i$uadad !mites ate.aes

1@=51


14/21


( ) ( ) =+ 100 2xlm 1 ( ) ( ) =+ + baxlm 00 b 1=b

( ) ( ) =+ baxlm 33 13 +a ( ) ( ) == ++ 533xlm 2 1=a

#$%58 ' 64 Los t.oos de*inito.ios son :c!ti!u#< Ha" 7ue i$uaa. !mites ate.aes ( ) 0xlm 3 ( ) ( ) =+ ++ nmxlm 00 n 3=n ( ) ( ) =+ nmxlm 22 32 +m ( ) +2xlm 1 2=m

#$%53 ' 65 a( ( )( )

+

+++

+23

12

213

1121

n

n

n

naa

nn ( )( ) >

++ 02353

7

nn nn aa >+1 $recie!te

+

23

12

n

nan < 1na

( )( ) ( )

ce0to en { }= 12963t

#$%511 ' 84 =

++

x

x 34

21

++

++ 342

2

34

34

21

x

x

x

x =42

e21e

=

+

+

2

24

11

ax

x

( )

++

+

+

2

2

24

1

1

4

24

11

x

ax

x

x

( )

4

1 a

e

( )

=4

1

2

1 a ( )12 = a

#$%511 ' 85 ( )

( )!1

12 11

1

+

+=

++

+

n

na

nn

n

( )

( )

=

+

+=

+++

!12

!12 11

1

nn

nn

a

ann

nn

n

n ( )

( )

=

+

+ +

1

12 1

nn

nn

n n

n

n

+ 1

2

n

n

n

na

a

+=+

1121 ( ) =

+

n

n

a

axlm 1 e2

#$%511 ' 86a( ( ) xf .a m/s estrecha ( ) xg .a m/s ancha )( ( ) ( )xgxf cuando x

c( ( ) ( ) ( )( ) ( )++ 32227 23 xxxxxxgxf ( )28 +x( )[ ]=+ gfxlm 0 ( )[ ]= gfxlm 0

d( ( ) ( ) 28

322 ++ xxxxf 322 xx, ( )xg 8R

x ( ) ( ),lmflm = ( ) ( )glmflm ( ) ( )xg%sntotaxf

#$%511 ' 87 a( La $.*ica .e0.esenta a recta ( ) a 3= xyCoo.denadas en o.i$en ( )03 " ( )30 #endiente 1=m O.denada o.i$en 3=n

)( ( )b ( ) ( )

+

+

= 333

x

xx

y 3= xy#a.a 3=x a *uncin est :i!"etermi!a"a

Unidad 8 - Funciones, Límites y Continuidad - Problemas resueltos

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Transcript of Unidad 8 - Funciones, Límites y Continuidad - Problemas resueltos