1. =
−
+−
→ ee
xx
xx
ln1
lim
2
1 ee
x
x
xx
3
1
2
lim
1
=
+
→
1)

2. 1
1
lim
1
1
lim
ln
1
lim
111
-=
-
=
-
=
- x
x
x
x
xxx →→
0
0
LH→
2)

3. ( )
( )
0
1
0
0coscos
lim
)cos-1(
coscos-1
lim
cos-
coscos-1
lim
-
cos
cos-1
lim
-tan
cos-1
lim
0→0→
0→
0→0→
====
==
==
sensenx
x
xsenx
xx
xsenxsenx
xx
senx
x
senx
x
senxx
x
xx
x
xx
3)

4. 2
1
2lim
2
2
lim
22
22
lim
2
82
lim
1
1
11
===
==
+
++
-
∞→
∞
∞
∞→
∞→
∞
∞
∞→
-
x
x
x
x
x
x
x
x
x
x L
L
4)

5. 2
1
2
cos
limlim
1
2
lim
2
lim
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
→→
→→
5)

6. 0
1
2
cos
2
lim
1-
1
2
lim
1→
0
0
1→
=






=
−




 x
x
x
sen
xLHx
πππ
6)

7. ( ) ( )
( )
( ) 8
1
)(cos248
lim
)(24
cos
lim
)2(22
cos
1
lim
2
)ln(
lim
2
0
0
2
2
0
0
2
2
-
=
-
-
=
=
-
=
=
-
=
-
-
--
→
→
→→
xxπsenx
senx
senxxπ
x
xπ
x
senx
xπ
senx
π
x
LH
π
x
OPERO
π
x
LHπ
x
7)

8. 0
1
0cos2
lim
cos
lim
1
cos
1
lim
1
)ln(
lim)ln(lim
2
0
0
0
2
0
2
00
)·(0
0
=
-
=
cos-
-
=
-
=
=
-
==
++
+++
∞
∞
∞-
x
senxxxx
senx
xx
x
x
senx
x
senx
senxx
xLHx
opero
xLHxoperox
→→
→→→
8)

9. 0
1
0
cos
)(cos
lim
coscos
lim
cos
)cos1(limcot)cos1(lim
0
0
0
2
0
0
·0
0
=
=
-2--
=
-
=
=-=-
x
senxxsenx
senx
xx
senx
x
xgxx
xLHx
OPERO
xINDx
→→
→
∞
→
9)

10. ( )
( )
=
+
=
-
+
=
=
-+
-+
=
=
-
+--
=-
-
2
1
11
1
lim
1
ln
ln
lim
1
·1ln
11··ln1
lim
ln1
1ln
lim
ln
1
1
lim
2
1
0
0
1
1
0
0
11
xx
x
x
x
x
x
x
xx
xxx
xx
xxx
xx
x
xLHx
OPERACIONS
ARRANXOx
LHx
OPERO
INDx
∞∞
→→
→
→→
10)

11. ∞
→
=
·0
22
0
lnlim xx
x
∞
∞
→
=
HLx
x
x
´
2
2
0 1
ln
lim
=
−
→
4
2
0 2
2
lim
x
x
x
x
x
0lim 2
0
=−
→
x
x
=
11)

12. ( )
( )
( )
( )
( )
( ) ( )( )
( )
ππ
πxπsen
πxπxxπsen
xπ
xπsenx
xπ
xπsen
xxπtagx
x
LHx
opero
x
def
x
2
)2/(
1
)2/(2/
)2/(2/cos)1(2/1
lim
2/cos
2/)1(
lim
2/cos
2/
)1(lim2/)1(lim
1
0
0
1
11
=
-
-
=
=
-
-+-
=
=
-
=
=-=-
→
→
→→
12)

13. 0
1
0
1
ln2ln1
lim
1
1
·ln2ln1
lim
1
ln
lim
ln
1
1
1
1
lim
ln
1
)1ln(
lim)1·ln(lnlim
2
1
2
1
0
0
2
1
2
11
·0
1
=
-
=
-
+
=
-
+
=
+-
=
=
-
-=
-
=-
+
++
+++
∞
∞
∞
xx
x
xxx
x
xx
x
x
x
x
x
xx
x
simplifico
xLHx
opero
xLHxx
→
→→
→→→
13)

14. 30
lim
x
xsenx
x
−
→
=
−
→ 20 3
cos1
lim
x
x
x
6
1
6
cos
lim
6
lim
6
)(
lim
000
===
−−
→→→
x
x
xsen
x
xsen
xxx
=
14)

15. 15)

16. 1
0
1
0
cos1
cos2
lim
0
0
cos
lim
cos
1
lim
1
ln
lim
·0
·lnlimlnlimlimlnln
lim
0
0
2
0
2
0
0000
0
==
=
-
=
+-
=
-
=
-
=
=====
=
eA
xsenxx
xsenx
xx
xsen
xsen
x
x
senx
x
xsenxxxA
Ax
xLHx
opero
x
LHxINDx
senx
x
senx
x
senx
x
→→→
→→→→
→
∞
∞
∞
16)

17. ( )
2
1
2
2
)12)(2(2
·
2
2
1
lim
2
2
ln
lim
2
2
lnlim
2
2
lnlim
2
2
limlnln
2
2
lim
22
2
2
0→
0
0
2
2
0→2
1
0→
1
20→
1
20→
2
11
20→
2
22
2
−=
++
++−++
++
+
=






++
+
=





++
+
=
=





++
+
=





++
+
=
==





++
+ −
x
xx
xxxx
xx
x
x
xx
x
xx
x
xx
x
xx
x
A
eA
xx
x
x
LHx
x
x
x
x
x
x
x
x

18. ( )
( ) ( )
( )
( )
0
1
02
lim
1
lim
1
1
1
lim
1
1ln
lim1lnlim
1lnlim1limlnln
11lim
2
0
2
0
2
00
·0
0
log0lim0
0
=
-
=
-
+
=
+-
=
-
-
=
-
=-=
=-=-=
==-
x
xx
xLH
x
x
x
ARRANXO
OPERAC
x
x
xLH
x
xind
x
x
spropiedade
xx
x
spropiedade
xx
x
xx
x
e
exxe
e
ex
x
e
e
x
e
ex
eeA
Ae
→
0
0
→
→
∞
∞
→
∞
→
→→
→
18)

19. 1
0
1
1
lim
ln
limln
1
lim
lnlimlimlnln
lim
0
log
1
lim
1
1
==
====
===
=
eA
x
x
x
x
x
xxA
Ax
xLHx
opero
x
propi
x
x
propied
x
x
x
x
∞→
∞
∞
∞→∞→
∞→∞→
∞→
19)

20. ( )
( ) ( )
1
2
0
0
0
0
2
0
2
0
2
0
0
ln
1
0
ln
1
0
ln
1
0
1
1
lim
cos
lim
coslimlim
1
1
lim
ln
cotln
limcotlnlimcotlimlnln
cotlim
-
2
=
-=
+xcos-
=
-
=
=
-
=
-
=-
====
=
eA
xsenxxsen
x
xsen
x
senx
x
xsen
xtagx
x
xsen
tagx
x
gx
gxgxA
Agx
xLHx
simplifico
x
opero
x
opero
x
LHx
x
x
x
x
x
x
→→
→→→
→→→
→
∞
∞
20)

21. eeA
x
x
x
xxA
Ax
xLHx
x
x
x
x
x
x
==
==
-
=
===
=
--
-
1
1
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
lim
1
ln
lim
lnlimlimlnln
lim
→→
→→
→
21)

22. ( )
( ) ( )
( )
0
1
coscot
1
lim
cos1
1
·
cot
1
lim
1
cotln
lim
·0
cotlnlim
cotlnlimcotlimlnln
1cotlim
0
2
2
0
00
00
0
0
===
-
-
=
===
===
===
∞
∞
∞
∞
→→
→→
→→
→
xax
xsen
x
xsenax
senx
ax
gxsenx
gxgxA
eAgx
x
simplifico
x
LHxINDx
senx
x
senx
x
senx
x
22)

23. ∞=
∞
==
+
=
+
=
+
=
+
∞
∞
∞
∞∞
∞
∞
∞
11
2
lim
)1(
2
lim
)1(
2
lim
2
lim
1
lim
0→→
→
2
→
2
→
x
xLH
x
x
x
xx
xsimplificoxx
x
xLHx
x
x
e
x
e
xe
ee
xee
e
xe
e
23)

24. 24)
( )
( ) ( ) 2
2
4
2cos22
cos2
lim
2cos
2
lim
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. 25)
3
4
2
1
2
2
2
1
2
2
12
1
lim
-
)12ln(
lim
1→
0
0
21→
=
−
=
−
−=
−
x
x
x
xx
x
xLHx

26. 26)
0
1
2
1
1
lim
)1ln(
lim
2
0→
0
0
2
0→
=+=
+
x
x
x
x
xLHx

27. 5
3
5cos555
3cos333
lim
5)·5(5cos6
3)·3(3cos10
lim
5cos3
3cos5
lim
3·
3cos
1
5·
5cos
1
lim
3
5
lim
22
22
2
2
2
2
22
2
22
=
-
-
=
=
-
-
===
xxsen
xxsen
xsenx
xsenx
x
x
x
x
xtag
xtag
π
x
LHπ
x
LHπ
x
π
x
LHπ
x
→
0
0
→
0
0
→→
∞
∞
→
27)

28. 1
2/1·1·2
1
cos22
1
lim
22
cos
1
lim
2cos
1
lim
2
4
2
4
0
0
4
===
=
-
-
=
-
xxsen
xsen
x
x
tagx
π
x
OPERO
π
x
LHπ
x
→
→→
28)

29. 2
1
)(coscos2
1
lim
)(coscos2
lim
cos
)cos1(
lim
cos
)cos1(
lim
cos
cos
limcoslimlim
0
2
0
0
0
2
0
3
0
3
0
3
0
3
0
=
-+
=
=
-+
=
-
=
-
=
-
=
-
=
-
senxsenxxx
senxxsenxxsenx
senx
xxsen
x
xxsen
xsenx
xsen
x
xsenxsenx
xsen
senx
x
senx
xsen
senxtagx
x
simplifico
x
LHx
simplifico
x
opero
x
opero
x
sust
x
→
→
→→
→
→→
29)

30. 1
2
2
2
cos22
lim
cos2
2
lim
cos2cos
lim
cos2cos
lim
cos2cot
lim
2
0
0
2
2
22
-=
-
=
-
+
=
-
=
=-=
=-=-
senx
xxsenx
x
πxsenx
x
π
x
xsenx
x
π
senx
x
x
x
π
agx
x
π
x
LHπ
x
OPERO
π
x
OPERO
π
x
π
x
→→
→
→
DEF
→
30)