limites de funciones i

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


2/8


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

+ = = = = + + + ++ +


3/8


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


4/8


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

++

+ = = =


5/8


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

+ += = = =+ +


6/8


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

+ + +

+

= = = = +

+ = = +

= = = = + + +


7/8


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

+ +

= == =

= =


8/8


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

+ ++ + + ++ +

+ += = = = =

limites de funciones i

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