52338-Exercicios Sobre Limites (2)
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Transcript of 52338-Exercicios Sobre Limites (2)
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8/11/2019 52338-Exercicios Sobre Limites (2)
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EXERCICIOS SOBRE LIMITES:
Calcule os seguintes limites.
1)Soluo:
4
!"
#"
lim)!")$!"$
)#")$!"$
lim1%"
1%"&"
lim !"!"!
!
!" =+
=+
=
+
!)Soluo:
3)Soluo:
4)Soluo:
#)Soluo:
6)Soluo:
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=
++
#"
&"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
.
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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
it -ollo+s -om t,e Suee>e /incile t,at
.
1D) Calcule"!
"cos!lim"
SOLUTION :Aist note t,at
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
it -ollo+s -om t,e Suee>e /incile t,at
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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
it -ollo+s -om t,e Suee>e /incile t,at
.!%) 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!$"lim
!!
" ++
igual a =
+
%1
Similal2
1%%""lim
!
" += .
T,us it -ollo+s -om t,e Suee>e /incile t,at
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1%%"
)"sin!$"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
SOLUTION :Aist note t,at
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
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.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
= % .
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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
$Cicum7ent t,is in*eteminate -om 82 -actoing t,e numeato an* t,e *enominato.)
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$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
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-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
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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! .
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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"
+==
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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*.