Download - Limites de Funciones I
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8/7/2019 Limites de Funciones I
1/8
CEMATH LMITES DE FUNCIONES
Pgina 1
LMITES DE FUNCIONES
Ejercicio 1
Calcular el
2 22 2lim
1 1x x x x
x x+
+
Solucin
( )( ) ( )( )( )
2 22 2 2 1 2 12 2lim lim
1 1 1 1x x
x x x x x x x x
x x x x + +
+ = = = + +
3 2 2 3 2 2
2 2
2 2 2 2 5lim lim 51 1x x
x x x x x x x x
x x+ +
+ += =
Ejercicio 2
Calcular el
4 3
22
4lim
4x x x x
x
Solucin
( )( )( )( )
( )( )
2 24 3
22 2 2
2 2 24 0 2 8lim lim lim 4
4 0 2 2 2 4 x x x
x x x x x x x x x x
x x x x
+ + + + = = = = =
+ +
Ejercicio 3
Calcular el ( )2 2lim 5 9 6 8x
x x x x
+ +
Solucin
( )( )( )
( )
2 2 2 2
2 2
2 2
5 9 6 8 5 9 6 8
lim 5 9 6 8 lim5 9 6 8
x x
x x x x x x x x
x x x x x x x x
+ +
+ + + + +
+ + = = =+ + +
( )
( ) ( )
2 2
2 2 2 2
5 9 6 8 11 17 11lim lim
25 9 6 8 5 9 6 8x x
x x x x x
x x x x x x x x+ +
+ + = =
+ + + + + +
Ejercicio 4
Calcula0
( ) ( )lim ( )h
f x h f xsiendo f x x
h
+ =
Solucin
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8/7/2019 Limites de Funciones I
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CEMATH LMITES DE FUNCIONES
Pgina 2
( )( )
( )0 0 0( ) ( ) 0
lim lim lim0h h h
x h x x h xf x h f x x h x
h h h x h x
+ + ++ + = = = =
+ +
( ) ( )0 01lim lim
2h h x h x h
xh x h x h x h x
+ = =
+ + + +
Ejercicio 5
Calcular el23
3lim
4 3x
x
x x
+
Solucin
( )( )
23 3
3 0 3 1lim lim
0 3 1 24 3x x
x x
x xx x
= = =
+
Ejercicio 6
Calcular el
( )
3 2
3
2 3 5lim
1 3 2 2x
x x
x x
+
+
Solucin
( )
( )3 23
2 1 32 3 5 2 6 2
lim 2 21 31 3 2 2x
x x
x x
+ = = =
++
Ejercicio 7
Calcular
33
22
2 5lim
4 2x
x x
x x
+
+
Solucin
3 33 33 3
2 22 2
2 5 2 5 9 3 27lim lim
16 4 644 2 4 2x x x x x x
x x x x + + = = = =
+ +
Ejercicio 8
Calcula el valor de m en3 2
3 2
4 4( )
3 2 2
x x xf x
x mx x
+=
+ + , sabiendo que,
2
3lim ( )
7xf x
=
Solucin3 2
3 22
4 4 6 6 6 3lim 1024 4 4 2 4 26 4 26 73 2 2x
x x x mm m m x mx x
+ = = = = + + + ++ +
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8/7/2019 Limites de Funciones I
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CEMATH LMITES DE FUNCIONES
Pgina 3
Ejercicio 9
Calcula
2
22
6lim
4x
x kx
x
+
, sabiendo que dicho lmite es finito.
Solucin
( )( )
( )( )
2
22 2
2 36 4 2 6 5lim 2 2 0 1 lim
0 2 2 44x x
x x x kx k k k
x xx
++ + = = = =
+
Ejercicio 10
Calcular
3
41
1lim
1x
x
x
+
Solucin
( )( )
( )( )( ) ( )( )
23 2
4 2 21 1 1
1 11 1 3lim lim lim
41 1 1 1 1 1 x x x
x x x x x x
x x x x x x
+ ++ += = =
+ + +
Ejercicio 11
Calcular1 1
limx
x x
x
+
Solucin
( ) ( )
( ) ( )
1 1 1 11 1 1 1 2lim lim lim 0
1 1 1 1 x x x
x x x x x x x x
x x x x x x x
+ + + + + += = = =
++ + + +
Ejercicio 12
Calcular0
limt
x t x
t
+
Solucin
( )( )
( ) ( )0 0 0 01 1
lim lim lim lim2t t t t
x t x x t x x t x x t x
t x t x xt x t x t x t x
+ + ++ + = = = =
+ ++ + + +
Ejercicio 13
Calcula
2 3
2
3 2 1 5lim
3 2 2 3x x x x x
x x x+
+ +
+ + +
Solucin
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8/7/2019 Limites de Funciones I
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CEMATH LMITES DE FUNCIONES
Pgina 4
( )( ) ( )( )
( )( )
2 2 32 3
2 2
3 2 1 2 3 5 3 23 2 1 5lim lim
3 2 2 3 3 2 2 3x x
x x x x x x x x x x x
x x x x x x+ +
+ + + + + + + = = =
+ + + + + +
4 3 3 4 3
3
3 6 2 .... 3 2 ... 2lim
3 .... 3x x x x x x
x++ + +
=+
Ejercicio 14
Hallar
13
2
2 1lim
7 1
x
x
x
x
x
+
+
Solucin
1 33 2
2
2 1 7 7 7lim
7 1 15 15 15
xx
x
x
x
+
= =
+
Ejercicio 15
Calcular0
4 2limx
x
x
+
Solucin
( )( )( ) ( )0 0 0 0
4 2 4 24 2 0 4 4 1 1) lim lim lim lim
0 44 24 2 4 2 x x x x
x xx xa
x x x x x x
+ + ++ + = = = = =
+ ++ + + +
Ejercicio 16
Calcular4 2
limx
x
x
+
Solucin
4 2lim 0
x
x
x+
+ =
Ejercicio 17
Hallar
2
1
( 5)
5
4 2lim
x
x
x
x
+
Solucin
2
1 1( 5) 0
54 2 1 1lim 0
5 5
x
xx
x
++
+ = = =
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CEMATH LMITES DE FUNCIONES
Pgina 5
Ejercicio 18
Hallar0
( )lim
n n
h
x h x
h
+
Solucin
1 2 2
0 0
.......0 1 2( ) 0
lim lim0
n n n n n
n n
h h
n n n n x x h x h h x
n x h x
h h
+ + + +
+ = = =
1 2 2
1 2 1 1
0 0
.......1 2
lim lim .......2
n n n
n n n n
h h
n n n x h x h h
nnnx x h h nx
h
+ + +
= + + + =
Ejercicio 19
Calcular20
limx
x
x x +
Solucin
Sabemos que( )
( )
2
1si 0
1 1si 0
si 0 1 si 01 1
xx
x x xxx xx
x x xx x x x x x
= + +
0
20
0
1lim 1
1lim
1lim 1
1
x
x
x
x x
x x
x
+
= +
= + = +
+
Por tanto20
limx
x
x x
+
Ejercicio 20
Calcular1
1lim
1xx x x x
x
+
Solucin
( ) ( ) ( ) ( )( )( )( )( )1 1 1 1
1 1 1 1 1 1 11 0lim lim lim lim
1 0 1 1 1 1 x x x x
x x x x x x x x x x x x
x x x x x
+ + + + + = = = = =
+
( )( )( )( )1 1
1 1 1 2lim lim 1211 1
x x
x x xxx x
+ += = = =+ +
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CEMATH LMITES DE FUNCIONES
Pgina 6
Ejercicio 21
Calcular0
lim
x
x
xe
Solucin
1
0 0
0 01
0 0
1lim lim
lim lim
lim lim
x
xx x
x xx x
xx x
x
x x
e ee
e e
e e e
+ +
= =
=
= =
Ejercicio 22
Calcular ( )
2
1
0lim 1x
x x+
Solucin
( ) ( )
2
0
2 20
0
11 1
1lim
11 1lim
10 0 0 0lim
01 1lim 1 1 lim 1 lim 1 lim 1
1 1
x
x
x
xx x
x
xx x
x x x x
x
e e x e x
e ex x
+
+
= = + = = + = + = = + = = +
Ejercicio 23
Dada la funcin
2
3 2
12
4
1( ) 2 0
2
10
si xx
f x si xx x
si xx
<
= <
Calcular los siguientes lmites:
0 2 1lim ( ) lim ( ) lim ( ) lim ( ) lim ( )
x x x x xf x f x f x f x f x
Solucin
21lim ( ) lim 0
4x xf x
x = =
( )
( )( )
( )
( )3 2 2
0 0 0
0 0
0 0
1 1 1lim lim lim
2 2 0lim lim
1 1lim lim
0
x x x
x x
x x
f x x x x x
f x f x
f xx
+ +
+
+
= = = = +
+ += = +
= = = +
( )
( )( )( )
( )( )
( )2
2 2 2
2 2
3 2 22 2 2
1 1 1lim lim lim
2 24 4 0lim lim
1 1 1lim lim lim2 2 4 0
x x x
x x
x x x
f xx xx
f x f x
f x x x x x
+ + +
+
= = = = +
+ = = +
= = = = + + +
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8/7/2019 Limites de Funciones I
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CEMATH LMITES DE FUNCIONES
Pgina 7
3 21 1
1 1lim ( ) lim 1
2 1x xf x
x x = = =
+
1lim ( ) lim 0x x
f xx
= =
Ejercicio 24
Calcular el10
1lim
1x xe
Solucin
1 100
1 10 0
1 100
1 1 1 1lim 1
1 1 01 1 1 1lim
1 1 1 11 1 lim 01 11 1
x x
x x
x x
ee e
e e
ee e
++
+
= = = =
= =
= = = =
10
1lim
1x xe
Ejercicio 25
Calcular k para que el ( )3
lim ( ) 3x
f x f
= siendo
2 9si 3
( ) 3
si 3
xx
f x x
k x
=
=
Solucin
( ) ( )2
3 3 3
3 39lim ( ) lim lim 6
3 3 x x x
x xxf x
x x
+ = = =
( )3f k=
Luego, 6k=
Ejercicio 26
Dada la funcin2
0
( ) 1 0 1
1
xe si x
f x x si x
x si x
= <
> Calcular
0 1lim ( ) y lim ( )x x
f x f x
Solucin
Hallamos( )
0 0
20 0
0 0
lim ( ) lim 1lim ( ) lim ( ) 1
lim ( ) lim 1 1
x
x x
x x
x x
f x e f x f x
f x x
+ +
= == =
= =
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8/7/2019 Limites de Funciones I
8/8
CEMATH LMITES DE FUNCIONES
Pgina 8
Hallamos( )2
1 1
1
1 1
lim ( ) lim 1 0lim ( )
lim ( ) lim 1
x x
x
x x
f x xf x
f x x
+ +
= ==
= =1
lim ( )x
f x
Ejercicio 27
Calcular ( )3 3lim 1n
x x
Solucin
( )( )
( ) ( )
3 3
23 2 33
1 1lim 1 lim 0
1 1n n
x xx x
x x x x
= = =
+ +
Ejercicio 28
Calcular
32
2
2 1lim
x
x
x x
x
+
+ +
Solucin
( ) ( )2 2
2 2 2
2 13 2 1 2 7 32 lim 3 1 lim 3 lim2
2
2 1lim 1
x x x
x xx x x xx xx x x
x
x xe e e e
x
+ ++ + + ++ +
+ += = = = =