52338-exercicios sobre limites (2)

8/11/2019 52338-Exercicios Sobre Limites (2)

1/13

EXERCICIOS SOBRE LIMITES:

Calcule os seguintes limites.

1)Soluo:

4

!"

#"

lim)!")$!"$

)#")$!"$

lim1%"

1%"&"

lim !"!"!

!

!" =+

=+

=

+

!)Soluo:

3)Soluo:

4)Soluo:

#)Soluo:

6)Soluo:


2/13


3/13

=

++

#"

&"lim

"

>eoaten*em"

#

e"

&

"uan*oe

"

#

"

&1

lim

"

#

"

"

"

&

"

"

lim#"

&"

lim

teemoso "e"esso*a*enomina*oenumea*oo?i7i*in*o

""" +

+

=+

+

=+

+

?esta -oma#"

&"lim

" ++

ten*e a

1

14) Calcule4"1%"

&""lim

!

" ++

Soluo:

6ote t,at t,e e"ession "!5 " lea*s to t,e in*eteminate -om 0 asx se aoac,es .Cicum7ent t,is 82 *i7i*ing eac, o- t,e tems in t,e oiginal o8lem 82 " t,e ,ig,est o+e o-xin

t,e o8lem .

When x the

4"1%"

&""lim

!

" ++

aoac,es %

= 0

15) Calcule &"0"lim!

"+

Soluo:

&""

&""lim

&""

)&"")$&"0"$lim&"0"lim

!

!!

"!

!!

"

!

"

++

=

++

+++=+

=

&""

&lim

!" ++

e uan*o

" &"0"lim!

"+

=

&""

&lim

!" ++

ten*e a >eo

16) $Cicum7ent t,is in*eteminate -om 82 using t,e con@ugate o- t,e e"ession

Calcule"

)"sin$lim"

SOLUTION :Aist note t,at

8ecause o- t,e +ell0no+n oeties o- t,e sine -unction. Since +e ae comuting t,e

limit asxgoes to in-init2 it is easona8le to assume t,atx % . T,us

.


4/13

Since

it -ollo+s -om t,e Suee>e /incile t,at

1&) Calcule""cos!lim

" +

SOLUTION :

Aist note t,at

8ecause o- t,e +ell0no+n oeties o- t,e cosine -unction. 6o+ multil2 82 01 e7esingt,e ineualities an* getting

o

.

6e"t a** ! to eac, comonent to get.

Since +e ae comuting t,e limit asxgoes to in-init2 it is easona8le to assume t,at x3 %. T,us

.Since


.

1D) Calcule"!

"cos!lim"


8ecause o- t,e +ell0no+n oeties o- t,e cosine -unction an* t,ee-oe

.Since +e ae comuting t,e limit asxgoes to in-init2 it is easona8le to assume t,at 0 !x %. 6o+ *i7i*e eac, comonent 82 0 !x e7esing t,e ineualities an* getting

o

.Since



5/13

1') Calcule

"

!cos"lim

%"

SOLUTION :6ote t,at

"

!cos"lim

%"?OES 6OT EXIST since 7alues o-

!

"cos oscillate

8et+een 01 an* 31 asxaoac,es % -om t,e le-t. Fo+e7e t,is *oes 6OT necessail2 mean

t,at

"

!cos"lim

%"*oes not e"ist G H In*ee*x % an*

-ox %. Multil2 eac, comonent 82x e7esing t,e ineualities an* getting

o

.

Since


.!%) Calcule

.SOLUTION :Aist note t,at

so t,at

an*

.Since +e ae comuting t,e limit asxgoes to in-init2 it is easona8le to assume t,at

x31%% %. T,us *i7i*ing 82x31%% an* multil2ing 82x! +e get

an*

.T,en

"

1%%1

"!lim

"

1%%""

"!

lim1%%"

"!lim

"

!

"

!

" +=

+=

+ uan*o " ten*e a

1%%"

)"sin!$"lim

!!

" ++

igual a =

+

%1

Similal2

1%%""lim

!

" += .

T,us it -ollo+s -om t,e Suee>e /incile t,at


6/13

1%%"

)"sin!$"lim

!!

" ++

= $*oes not e"ist).

!1) Calcule

.SOLUTION :Aist note t,at

so t,at

an*

.T,en

= # .Similal2

#1%"

1"#lim

!

!

"=

++

T,us it -ollo+s -om t,e Suee>e /incile t,at

#1%"

)"sin$"#lim

!

!

"=

+

.

22) Calcule


an*

so t,at

an*

.

Since +e ae comuting t,e limit asxgoes to negati7e in-init2 it is easona8le toassume t,atx0 %. T,us *i7i*ing 82x0 +e get

o


7/13

.6o+ *i7i*e 82x!3 1 an* multil2 82x! getting

.

T,en

%%%%1

%

"

"

1

"

1

"

!

lim

!

"=

+=

+

Similal2

%)")$1"$

"!lim

!

!

"=

+

It -ollo+s -om t,e Suee>e /incile t,at

%)")$1"$

)"cos"$sin"lim

!

!

"=

++

LIMITES CONSTINUIDADE

1) ?etemine se a seguinte -uno contJnua emx=1 .

SOLUTION ::Aunctionfis *e-ine* atx = 1 sincei.)f$1) = ! .

T,e limit

= $1) 0 #

= 0! i.e.

ii.) !)"$-lim1"

= .

But

iii.) )1$-)"$-lim1"

so con*ition iii.) is not satis-ie* an* -unctionfis 6OT continuous at x = 1.

!) ?etemine se a seguinte -uno contJnua emx = 0! .

SOLUTION :Aunctionfis *e-ine* atx=0! since

f$0!) = $0!)!3 !$0!) = 404 = % .

T,e le-t0,an* limit

= $0!)!3 !$0!)= 4 0 4

= % .


8/13

T,e ig,t0,an* limit

= $0!)0 K$0!)

= 0D 3 1!= 4 .

Since t,e le-t0 an* ig,t0,an* limits ae not eual

ii.) )"$-lim!" *oes not e"ist

an* con*ition ii.) is not satis-ie*. T,us -unctionfis 6OT continuous atx = 0! .

) ?etemine se a seguinte -uno contJnua emx = % .

SOLUTION :Aunctionfis *e-ine* atx = % sincei.)f$%) = ! .

T,e le-t0,an* limit

= ! .T,e ig,t0,an* limit

= ! .

T,us )"$-lim%"

e"ists +it,

ii.) !)"$-lim%"

= .

Since

iii.) )%$-!)"$-lim%"

==

all t,ee con*itions ae satis-ie* an*fis continuous atx=% .

4) ?etemine se a -uno1"

1")"$,

!

++

= contJnua atx = 01 .

SOLUTION :Aunction his not *e-ine* atx = 01 since it lea*s to *i7ision 82 >eo. T,us , $01) *oes not e"ist con*ition i.) is 7iolate* an* -unction his 6OT continuous

atx= 01 .

#) C,ec t,e -ollo+ing -unction -o continuit2 atx = an*x = 0 .

SOLUTION :Aist c,ec -o continuit2 atx= . Aunctionfis *e-ine* atx= since

.T,e limit

$Cicum7ent t,is in*eteminate -om 82 -actoing t,e numeato an* t,e *enominato.)


9/13

$Recall t,atA!0B!= $A 0B)$A 3B) an*A0B= $A 0B)$A! 3AB 3B!) . )

$?i7i*e out a -acto o- $x 0 ) . )

i.e.

ii.) .Since

iii.) all t,ee con*itions ae satis-ie* an*fis continuous atx= . 6o+ c,ec -o continuit2 at

x = 0 . Aunctionfis not *e-ine* atx= 0 8ecause o- *i7ision 82 >eo. T,us

i.)f$0)*oes not e"ist con*ition i.) is 7iolate* an*fis 6OT continuous atx = 0 .

K) /aa ue 7aloes *exa -uno 4""

#"")"$-

!

!

+++= contJnua H

SOLUTION 6 :Aunctionsy=x!3 x3 # an*y=x!3 x0 4 ae continuous -o all7alues o-xsince 8ot, ae ol2nomials. T,us t,e uotient o- t,ese t+o -unctions

4""

#"")"$-

!

!

+++= is continuous -o all 7alues o-x+,ee t,e *enominato

y = x!3 x0 4 = $x 0 1)$x 3 4) *oes 6OT eual >eo. Since $x 0 1)$x 3 4) = % -ox = 1

an*x = 04 -unctionfis continuous -o all 7alues o-xEXCE/Tx = 1 an*x = 04 .

&) /aa ue 7aloes *exa -uno ( ) 1

!% )#"sin$)"$g += contJnua H

SOLUTION:Aist *esci8e -unctiongusing -unctional comosition. Letf$x) =x1 ,$") = sin$") an* k$x) =x!%3 # . Aunction kis continuous -o all 7alues o-xsince it is a

ol2nomial an* -unctionsfan* hae +ell0no+n to 8e continuous -o all 7alues o-x.T,us t,e -unctional comositions

an*

ae continuous -o all 7alues o-x. Since

-unctiongis continuous -o all 7alues o- x.

D) /aa ue 7aloes *exa -uno "!")"$- ! = contJnua H

SOLUTION :Aist *esci8e -unctionfusing -unctional comosition. Letg$x) =x!0 !x

an* ,$") = " . Aunctiongis continuous -o all 7alues o-xsince it is a ol2nomial an*

-unction his +ell0no+n to 8e continuous -o %" . Sinceg$x) =x!0 !x=x$x0!) it-ollo+s easil2 t,at %)"$g -o %" an* !" . T,us t,e -unctional comosition

is continuous -o %" an* !" an*. Since


10/13

-unctionfis continuous -o %" an* !" an*.

') /aa ue 7aloes *exa -uno

+

=!"

1"ln)"$- contJnua H

SOLUTION :Aist *esci8e -unctionfusing -unctional comosition. Let

!"

1")"$g

+

= an* ,$") = ln$"). Sincegis t,e uotient o- ol2nomialsy=x 0 1 an*

y= x 3 ! -unctiongis continuous -o all 7alues o-xEXCE/T +,eex3! = % i.e.

EXCE/T -ox= 0! . Aunction his +ell0no+n to 8e continuous -ox % . Since

!"

1")"$g

+

= it -ollo+s easil2 t,atg$x) % -ox 0! an*x 1 . T,us t,e -unctional

comosition

is continuous -ox 0! an*x 1 . Since

-unctionfis continuous -ox 0! an*x 1 .

1%) /aa ue 7aloes *exa -uno '"4

e)"$-

!

"sin

= contJnua H

SOLUTION 10 :Aist *esci8e -unctionfusing -unctional comosition. Letg$") = sin$") an* h$x) = ex 8ot, o- +,ic, ae +ell0no+n to 8e continuous -o all7alues o-x. T,us t,e numeato 2 = esin$")= ,$g$")) is continuous $t,e -unctional

comosition o- continuous -unctions) -o all 7alues o-x. 6o+ consi*e t,e *enominato

'"42 ! = . Letg$x) = 4 h$x) =x!0 ' an* ")"$B = . Aunctionsgan* h

9e continuous -o all 7alues o-xsince 8ot, ae ol2nomials an* it is +ell0no+n t,at-unction kis continuous -o %" . Since h$x) =x!0 ' = $x0)$x3) = % +,enx = o

x = 0 it -ollo+s easil2 t,at 0"an*-o "%)"$, -o an* so t,at

))"$,$B'"42 ! == is continuous $t,e -unctional comosition o- continuous

-unctions) -o 0"an*" an*. T,us t,e *enominato '"42 ! = iscontinuous $t,e *i--eence o- continuous -unctions) -o 0"an*" an*. T,eeis one ot,e imotant consi*eation. e must insue t,at t,e ?E6OMI69TOR IS

6ENER ERO. I-

t,en

.Suaing 8ot, si*es +e get

1K =x!0 'so t,at

x!= !#

+,enx= # ox= 0# .

T,us t,e *enominato is >eo i-x= # ox= 0# . Summai>ing t,e uotient o- t,ese

continuous -unctions'"4

e)"$-

!

"sin

= is continuous -o 0"an*"

an* 8ut 6OT -ox= # an*x= 0# .

/aa ue 7aloes *ex a seguinte -uno contJnua H


11/13

SOLUTION :Consi*e seaatel2 t,e t,ee comonent -unctions +,ic, *eteminef.

Aunction1"

1"2

= is continuous -ox 1 since it is t,e uotient o- continuous

-unctions an* t,e *enominato is ne7e >eo. Aunctiony= # 0xis continuous -o

1"! since it is a ol2nomial. Aunction4"

K2

= is continuous -ox 0! since it

is t,e uotient o- continuous -unctions an* t,e *enominato is ne7e >eo. 6o+ c,ec -ocontinuit2 o-f+,ee t,e t,ee comonents ae @oine* toget,e i.e. c,ec -o continuit2 at

x = 1 an*x = 0! . Aox= 1 -unctionfis *e-ine* since

i.)f$1) = # 0 $1) = ! .T,e ig,t0,an* limit

%

%

1"

1"lim)"$-lim

1"1"=

=++

$Cicum7ent t,is in*eteminate -om one o- t+o +a2s. Eit,e -acto t,e numeato as t,e*i--eence o- suaes o multil2 82 t,e con@ugate o- t,e *enominato o7e itsel-.)

= ! .

T,e le-t0,an* limit

== # 0 $1)

= ! .T,us

ii.) !)"$-lim1"

= .

Since

iii.) !)"$-lim1"

= = -$1)

all t,ee con*itions ae satis-ie* an* -unctionfis continuous atx = 1 . 6o+ c,ec -ocontinuit2 atx = 0! . Aunctionfis *e-ine* atx = 0! since

i.)f$0!) = # 0 $0!) = 11 .T,e ig,t0,an* limit

== # 0 $ 0!)

= 11 .T,e le-t0,an* limit

4"

Klim)"$-lim

!"!" =

=

= 01 .Since t,e le-t0 an* ig,t0,an* limits ae *i--eent

ii.) )"$-lim

!"

*oes 6OT e"ist

con*ition ii.) is 7iolate* an* -unctionfis 6OT continuous atx=0! . Summai>ing

-unctionfis continuous -o all 7alues o-xEXCE/Tx = 0! .


12/13

1!. ?etemine to*os os 7aloes *a constanteAaa ue a seguinte -uno se@a contJnua aa to*os os 7aloes*ex.

SOLUTION :Aist consi*e seaatel2 t,e t+o comonents +,ic, *etemine -unctionf.

Aunctiony=A!x0A is continuous -o " -o an2 7alue o-Asince it is a ol2nomial. Aunctiony= 4 is continuous -ox since it is a ol2nomial. 6o+ *etemineAso t,at -unctionfis continuous atx= . Aunctionfmust 8e *e-ine* atx = so

i.)f$)=A!$) 0A= A!0A.T,e ig,t0,an* limit

)9"9$lim)"$-lim!

""=

++

=A!$) 0A= A!0A.

T,e le-t0,an* limit

4lim)"$-lim""

= = 4 .

Ao t,e limit to e"ist t,e ig,t0 an* le-t0,an* limits must e"ist an* 8e eual. T,us

ii.) 499)"$-lim !

"

==

so t,at

= A!0A0 4 = % .Aactoing +e get

$A0 4)$A3 1) = %-o

49= oA= 01 .

Ao eit,e c,oice o-A

iii.)

all t,ee con*itions ae satis-ie* an*fis continuous atx = . T,ee-oe -unctionfis

continuous -o all 7alues o-xi- 4

9 = oA= 01 .

1. ?etemine to*os os 7aPoes *as constantes 9 e B aa ue a -uno se@a contJnua aa to*os os 7aloes *e

x.

SOLUTION :Aist consi*e seaatel2 t,e t,ee comonents +,ic, *etemine -unctionf.

Aunctiony=Ax0Bis continuous -o 1" -o an2 7alues o-Aan*Bsince it is aol2nomial. Aunctiony= !x!3 Ax3Bis continuous -o 1"1 -o an2 7alues o-A

an*Bsince it is a ol2nomial. Aunctiony= 4 is continuous -ox 1 since it is a ol2nomial.

6o+ *etemineAan*Bso t,at -unctionfis continuous atx=01 an*x=1 . Aist consi*econtinuit2 atx = 01 . Aunctionfmust 8e *e-ine* atx = 01 so

i.)f$01)=A$01) 0B= 0A0B.

T,e le-t0,an* limit

)B9"$lim)"$-lim1"1"

=

=

=A$01) 0B= 0A0B.

T,e ig,t0,an* limit

)B9""!$lim)"$-lim!

1"1"++=

++

=

= !$01)!3 A$01) 3B= ! 0 A3B.

Ao t,e limit to e"ist t,e ig,t0 an* le-t0,an* limits must e"ist an* 8e eual. T,us

ii.) B9!B9)"$-lim1"

+==


13/13

so t,at!A0 !B= !

o

$Euation 1)A0B= 1 .

6o+ consi*e continuit2 atx=1 . Aunctionfmust 8e *e-ine* atx=1 soi.)f$1)= !$1)!3 A$1) 3B= ! 3 A3B.

T,e le-t0,an* limit)B9""!$lim)"$-lim !

1"1"++=

=

= !$1)!3 A$1) 3B= ! 3 A3B.

T,e ig,t0,an* limit

4lim)"$-lim1"1"

++

= =

= 4 .Ao t,e limit to e"ist t,e ig,t0 an* le-t0,an* limits must e"ist an* 8e eual. T,us

ii.) 4B9!)"$-lim1"

=++=

o

$Euation !)A3B= ! .

6o+ sol7e Euations 1 an* ! simultaneousl2. T,usA0B= 1 an* A3B= !

ae eui7alent toA=B3 1 an* A3B= ! .

Qse t,e -ist euation to su8stitute into t,e secon* getting

$B3 1 ) 3B= ! B3 3B= !

an*4B= 01 .

T,us

an*

.Ao t,is c,oice o-Aan*Bit can easil2 8e s,o+n t,at

iii.) )1$-4)"$-lim1"

==

an*

iii.) )1$-!

1)"$-lim

1"=

=

so t,at all t,ee con*itions ae satis-ie* at 8ot,x=1 an*x=01 an* -unctionfis continuous at

8ot,x=1 an*x=01 . T,ee-oe -unctionfis continuous -o all 7alues o- xi-

401Ban*

49 == an*.

52338-exercicios sobre limites (2)

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