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=−+−
→ ee
xxxx
ln1lim
2
1 eex
x
xx
31
2lim
1=
+
→
1)
11
lim11
limln
1lim
111-=
-=
-=
- x
xx
xxxx →→
0
0
LH→
2)
( )
( )0
1
0
0coscoslim
)cos-1(
coscos-1lim
cos-
coscos-1lim
-cos
cos-1lim
-tan
cos-1lim
0→0→
0→
0→0→
====
==
==
sensenx
x
xsenx
xx
xsenxsenx
xx
senxx
senxx
senxx
x
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3)
2
12lim
2
2lim
22
22lim
2
82lim
11
11
===
==
+
++
-
∞→
∞
∞
∞→
∞→
∞
∞
∞→
-
xx
x
x
x
x
xx
x
x L
L
4)
21
2
coslimlim
1
2lim
2lim
0
0
0
0
0
0
0
0
0
0
==+
=-
=
=cos-
-+=
-
--
--
--
x
ee
senx
ee
x
ee
senxx
xee
xx
xLH
xx
x
LH
xx
xLH
xx
x
→→
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5)
012
cos2lim
1-
12lim
1→
0
0
1→=
=−
x
x
xsen
xLHx
πππ
6)
( ) ( )
( )
( ) 8
1
)(cos248lim
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coslim
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cos1
lim2
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2
0
0
2
2
0
0
2
2
-=
-
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=-
=
=-
=
-
-
--
→
→
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xxπsenx
senx
senxxπ
x
xπ
xsenx
xπ
senx
πx
LH
πx
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7)
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0cos2lim
coslim
1
cos1
lim1)ln(
lim)ln(lim
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0
0
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0
2
00
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0
=-
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++
+++
∞
∞∞-
x
senxxxx
senx
xx
x
xsenx
x
senxsenxx
xLHx
opero
xLHxoperox
→→
→→→
8)
01
0
cos
)(coslim
coscoslim
cos)cos1(limcot)cos1(lim
0
0
02
0
0
·0
0
=
=-2--
=-
=
=-=-
x
senxxsenx
senx
xx
senx
xxgxx
xLHx
OPERO
xINDx
→→
→
∞
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9)
( )
( )
=+
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=
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-+=
=-
+--=-
-
2
111
1
lim1ln
lnlim
1·1ln
11··ln1lim
ln1
1lnlim
ln
1
1lim
2
1
0
0
1
1
0
0
11
xx
x
xx
x
x
xxx
xxx
xx
xxx
xx
x
xLHx
OPERACIONSARRANXOx
LHxOPEROINDx
∞∞
→→
→
→→
10)
∞
→=·0
22
0lnlim xx
x
∞∞
→=HLx
x
x´
2
2
0 1ln
lim
=−
→
4
2
0 2
2
lim
x
xx
x
x0lim 2
0=−
→x
x
=
11)
( )( )( )
( )( )
( ) ( )( )( )
ππ
πxπsen
πxπxxπsen
xπ
xπsenx
xπ
xπsenxxπtagx
x
LHx
opero
x
def
x
2
)2/(
1
)2/(2/
)2/(2/cos)1(2/1lim
2/cos
2/)1(lim
2/cos
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1
0
0
1
11
=-
-=
=-
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=-
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→
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12)
01
0
1
ln2ln1lim
1
1·ln2ln1
lim1
lnlim
ln
11
11
lim
ln1)1ln(
lim)1·ln(lnlim
2
1
2
1
0
02
1
2
11
·0
1
=-
=-
+
=-
+=
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=-
-=-
=-
+
++
+++
∞
∞∞
xx
xxxx
x
xx
xx
x
x
xxx
x
simplifico
xLHx
opero
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→
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13)
30lim
x
xsenxx
−→
=−→ 20 3
cos1lim
x
xx
6
1
6
coslim
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→→→
x
x
xsen
x
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=
14)
15)
1
01
0
cos1
cos2lim
0
0
coslimcos
1
lim
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lim·0
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lim
0
0
2
0
2
0
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==
=-
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=-
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eA
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xx
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xsenx
x
senx
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xLHx
opero
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senx
x
senx
x
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x
→→→
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→
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16)
( )2
1
2
2
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221
lim
22
lnlim
2
2lnlim
2
2lnlim
2
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2
2lim
22
2
2
0→
0
0
2
2
0→2
1
0→
1
20→
1
20→
2
11
20→
2
22
2
−=++
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=
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x
xx
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x
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x
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x
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xA
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x
x
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x
x
x
x
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( )
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01
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1lim
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1
lim11ln
lim1lnlim
1lnlim1limlnln
11lim
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0
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00
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0
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=-
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==-
x
xx
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x
x
ARRANXO
OPERAC
xx
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x
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x
x
spropiedadexx
x
spropiedadexx
x
xx
x
e
exxe
e
ex
x
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x
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Ae
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0
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∞
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→
∞
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→→
→18)
1
01
1
limln
limln1
lim
lnlimlimlnln
lim
0
log
1
lim
1
1
==
====
===
=
eA
xx
xx
x
xxA
Ax
xLHx
opero
x
propix
x
propiedx
x
x
x
∞→
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∞
∞→∞→
∞→∞→
∞→
19)
( )
( ) ( )
1
20
0
0
0
20
20
2
0
0
ln1
0
ln1
0
ln1
0
11
limcos
lim
coslimlim1
1
lim
ln
cotlnlimcotlnlimcotlimlnln
cotlim
-
2
=
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=-
=
=-
=-
=-
====
=
eA
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x
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senxx
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Agx
xLHx
simplifico
x
opero
x
opero
x
LHx
x
x
x
x
x
x
→→
→→→
→→→
→
∞
∞
20)
eeA
xx
x
xxA
Ax
xLHx
x
x
x
x
x
x
==
==-
=
===
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--
-
1
1
0
0
1
11
1
11
1
11
1
11
1
lim1
lnlim
lnlimlimlnln
lim
→→
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21)
( )
( ) ( )
( )
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coscot
1limcos1
1·
cot1
lim
1cotln
lim·0
cotlnlim
cotlnlimcotlimlnln
1cotlim
0
2
2
0
00
00
0
0
===-
-
=
===
===
===
∞
∞
∞∞
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→
xaxxsenxxsenax
senx
axgxsenx
gxgxA
eAgx
x
simplifico
x
LHxINDx
senx
x
senx
x
senx
x22)
∞=∞==+
=+
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∞∞
∞
∞∞
∞∞
∞
11
2lim
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2lim
)1(
2lim
2lim
1lim
0→→
→
2
→
2
→
x
xLH
x
x
x
xx
xsimplificoxx
x
xLHx
x
x
e
x
e
xe
ee
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e
xe
e
23)
24)
( )
( ) ( ) 22
4
2cos22
cos2lim
2cos
2lim
cos2-lim
22
-
0→
0
0
2
-
0→
0
0
2
-
0→
==+−
++=
=+−=+
xxxsenx
xee
xx
senxee
senx
xee
xx
x
LH
xx
xLH
xx
x
25)
3
4
21
2
2
2
12
212
1
lim-
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1→
0
0
21→=
−=
−−=−
xx
xxx
xxLHx
26)
01
211
lim)1ln(
lim2
0→
0
02
0→=+=+ x
xx
xxLHx
5
3
5cos555
3cos333lim
5)·5(5cos6
3)·3(3cos10lim
5cos3
3cos5lim
3·3cos1
5·5cos1
lim3
5lim
22
22
2
2
2
2
22
2
22
=-
-=
=-
-
===
xxsen
xxsen
xsenx
xsenx
x
x
x
xxtag
xtag
πx
LHπx
LHπx
πx
LHπx
→
0
0
→
0
0
→→
∞
∞
→
27)
12/1·1·2
1
cos22
1lim
22cos1
lim2cos
1lim
2
4
2
4
0
0
4
===
=-
-
=-
xxsen
xsenx
x
tagx
πx
OPERO
πx
LHπx
→
→→
28)
2
1
)(coscos2
1lim
)(coscos2lim
cos
)cos1(lim
cos
)cos1(lim
coscos
limcoslimlim
0
20
0
0
20
30
3
0
30
30
=-+
=
=-+
=-
=-
=
-
=-
=-
senxsenxxx
senxxsenxxsenx
senx
xxsen
x
xxsen
xsenx
xsenx
xsenxsenx
xsen
senxx
senx
xsen
senxtagx
x
simplifico
x
LHx
simplifico
x
opero
x
opero
x
sust
x
→
→
→→
→→→
29)
12
2
2
cos22lim
cos2
2lim
cos2coslim
cos2coslimcos2cot
lim
2
0
0
2
2
22
-=-
=-
+=
-=
=-=
=-=-
senx
xxsenx
x
πxsenx
x
π
x
xsenx
x
π
senxxx
x
π
agx
x
πx
LHπx
OPERO
πx
OPERO
πx
πx
→→
→
→
DEF
→
30)
