problemas-resueltos-limites

LIMITES

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Erving Quintero Gil Ing. Electromecnico Bucaramanga Colombia 2010

1

Problema 1

lim y

(y + 1)2y2 +1

= lim y

y 2 + 2y + 1 y2 +1y2

Dividiendo la expresin por el mayor exponente que es y 2 2y 1 + + 2 y2 y2 y lim y y2 1 + y2 y2 Simplificando trminos semejantes

1+ lim y

2 1 + y y2 1+ 0 + 0 1 = = =1 1 1+ 0 1 1+ y2 y2 +1 =1

lim y

(y + 1)2

Problema 2

lim x

1000x x 2 -1

Dividiendo la expresin por el mayor exponente que es x2 1000x x2 lim x x2 1 x2 x2 Simplificando trminos semejantes

1000 x = 0 = 0 =0 lim x 1 1- 0 1 12 x lim x Problema 3lim x x 2 - 5x + 1 3x + 7

1000x =0 x 2 -1

2

Dividiendo la expresin por el mayor exponente que es x2 1 x 2 5x + 2 x2 x2 lim x x 3x 7 + x2 x2 Simplificando trminos semejantes1lim x 5 1 + x x2 1- 0 + 0 1 = = = 3 7 0+0 0 + x x2

lim x

x 2 - 5x + 1 = (No existe el lmite) 3x + 7

Problema 4

lim x

2x 2 - x + 3 x 3 - 8x + 5

Dividiendo la expresin por el mayor exponente que es x3 2x 2 x 3 + 3 x3 x3 lim x x x 3 8x 5 + x3 x3 x3 Simplificando trminos semejantes2 1 3 + x x2 x3 0+0+0 0 = lim x = =0 8 5 1+ 0 + 0 1 + 1x 2 x3 2x 2 - x + 3 lim x =0 x 3 - 8x + 5

Problema 5lim y

(2x + 3)3 (3x - 2)2x5 + 5

lim y

(2x + 3)3 (3x - 2)2x5 + 5

(2x )3 + 3(2x )2 (3) + 3(2x )(3)2 + (3)3 (3x - 2 )2 = lim y 5 +5 x

3

lim y

8x 3 + 3 4x 2 (3) + 3(2x ) 9 + 27 (3x - 2 )2 = 5 +5 x

8x 3 - 36x 2 + 54x + 27 9 x 2 - 12x + 4 5 +5 x

lim y

72 x 5 + 324x 4 + 486 x 3 + 243 x 2 - 96 x 4 - (36)(12)x 3 - 54(12)x 2 - 27(12)x = x5 + 5

Simplificando trminos semejantes

lim y

72 x 5 + 228 x 4 + 86 x 3 - 261 x 2 - 108 x + 108 = x5 + 5

Dividiendo la expresin por el mayor exponente que es

x5

72 x 5 228 x 4 86 x 3 + + x5 x5 x5 lim y x5 + x5Simplificando trminos semejantes72 + lim y

-

261 x 2 108 x 108 + x5 x5 x5

5 x5

lim y

(2x + 3)3 (3x - 2)2x5 + 5

228 86 261 108 108 + + x x2 x3 x 4 x 5 = 72 + 0 + 0 - 0 + 0 = 72 = 72 5 1 1+ 0 1+ 5 x= 72

Problema 6

lim x

2x 2 - 3x - 4 x4 +1

Dividiendo la expresin por el mayor exponente que es x2 2x 2 3x 4 2x 2 3x 4 2x 2 3x 4 x2 x2 x2 = x2 x2 x2 = x2 x2 x2 lim x x4 +1 x4 +1 x4 1 + x2 x4 x4 x4 Simplificando trminos semejantes

4

2 lim x

3 4 x x2 2-0-0 2 2 = = = =2 1 1 1 1+ 0 1+ x4 =2 x4 +1

lim x

2x 2 - 3x - 4

Problema 7lim x 2x + 3 x+ 3x

Dividiendo la expresin por el mayor exponente que es x 2x 3 2x 3 + + x x = x x lim x x x x 3x +3 + x x x x3 Simplificando trminos semejantes3 x = 2 + 0 = 2 = 2 =2 lim x 1 1+ 3 0 1+ 0 1 1+ 3 x2 2 +lim x 2x + 3 =2 x+ 3x

Problema 8lim x x2 10 + x x

Dividiendo la expresin por el mayor exponente que es x2 x2 2x 2

lim x

x2 x2 = 10 + x x 10 x x + x2 x2 x2


lim x

1 10 x + x x2

=

1 10 + x2 x x2

=

1 10 + x2 1 x

=

1 0+ 0

=

1 = 0

5

lim x

x2 10 + x x

= (No existe el lmite)

Problema 9

lim x

3 x2 +1 x +1

Dividiendo la expresin por el mayor exponente que es x 2 3 2 3 2 x +1 x +1 3 x +1 3 x + 1 x3 x3 x3 x x lim x = = = x +1 x 1 x 1 x 1 + + + x x x x x x x Simplificando trminos semejantes3 lim x 1 1 + 30+ 0 0 2 x3 x = =0 = 1 1 1+ 0 1+ x

lim x

3 x2 +1 x +1

= 0

Problema 10

x+ x + x Elevando la expresin al cuadrado. 2 x x lim x = 2 x+ x+ x x + x+ x Dividiendo la expresin por el mayor exponente que es x x x x x x x lim x = = = x+ x + x x x + x x x + x + + x x x x x2

lim x

x

( )

x x x + x x x + 2 x2 x

= x + x

x x x + 2 x x x4


lim x 1+

1 1 1 + x x3

=

1 1+ 0 + 0

=

1 1+ 0 + 0

=

1 1+ 0

=

1 =1 1

6

lim x

x x+ x + x

=1

Problema 11

lim x - 1

x3 +1 x2 + 1

Reemplazando el valor de (-1)

=

(- 1)3 + 1 = (- 1)2 + 1

-1 + 1 0 = =0 1+1 2

lim x - 1

x3 +1 = 0 2+1 x

Problema 12

lim x 5

x 2 - 5x + 10 x 2 - 25

Reemplazando el valor de (5)

=

(5)2 - 5(5) + 10 = 25 - 25 + 10 = 10 25 - 25 0 (5)2 - 25

=

lim x 5

x 2 - 5x + 10 = (No existe el lmite) x 2 - 25

Problema 13

lim x - 1 lim x - 1

x 2 -1 x 2 + 3x + 2 x 2 -1 x 2 + 3x + 2

= lim x - 1

(x - 1)(x + 1) (x + 2)(x + 1)x2 - 1 = (x - 1) * (x + 1) Factorizacion dos nmeros sumados sea 3 y multiplicados sea 2 x2 + 3x + 2 = (x + 2) (x + 1)

Simplificando trminos semejantes+ 2) Reemplazando el valor de (-1) = lim x - 1

(x

(x - 1)

((- 1) - 1) = - 2 = - 2 ((- 1) + 2) 17

lim x - 1

x 2 -1 x 2 + 3x + 2

=-2

Problema 14

lim x 1

x 3 - 3x + 2 x 4 4x + 3

x 2 + x - 2 (x - 1) x 3 - 3x + 2 lim x 1 = lim x 1 4 - 4x + 3 x 3 + x 2 + x - 3 (x - 1) x lim x 1

(x 1)(x + 2)(x - 1) (x 1) x 2 + 2x + 3 (x - 1)

Simplificando trminos semejanteslim x 1

x2 3x + 2 = (x2 + x - 2)(x - 1) Factorizacion dos nmeros sumados sea 1 y multiplicados sea -2 (x2 + x - 2) = (x + 1) (x - 2) x2 3x + 2 = (x - 1) (x + 2) (x - 1) x4 4x + 3 = (x3 + x2 + x 3) (x 1) x4 4x + 3 = (x2 + 2x + 3) (x 1) (x 1)

(x + 2) x 2 + 2x + 3

Reemplazando el valor de (1)=

12

(1 + 2) + 2(1) + 3

=

3 3 1 = = 1+ 2 + 3 6 2

lim x 1

x 3 - 3x + 2 1 = x 4 4x + 3 2

Problema 15

lim x 2 lim x 2

x 2 - 2x x 2 4x + 4

(x )(x - 2) x 2 - 2x = 2 - 4x + 4 (x - 2)(x - 2) x(x )

x2 2x = (x)(x - 2) Factorizacion dos nmeros sumados sea - 4 y multiplicados sea +4 (x2 - 4x + 4) = (x - 2) (x - 2)


( x - 2)


=

2 2 = = 2-2 0

8

lim x 2

x 2 - 2x = (No existe el lmite) x 2 4x + 4

Problema 16

lim x a lim x a

x 2 - (a + 1) x + a x3 a3 x 2 - ax - x + a x 2 - (a + 1) x + a = lim x a x3 - a3 x3 a3

lim x a

(x - 1)(x - a ) x 2 + ax + a 2 (x - a )

Simplificando trminos semejanteslim x a

x2 ax x + a ordenando para factorizar x2 x ax + a = x(x 1) - a(x 1) x2 x ax + a = (x 1) - (x a) x3 a3 = (x2 + ax + a2) ( x a)

(x - 1) x 2 + ax + a 2

Reemplazando el valor de (a)=

(a - 1) a 2 + a (a ) + a 2

=

a -1 a 1 = a2 + a2 + a2 3a2

lim x a

x 2 - (a + 1) x + a a - 1 = 3a 2 x3 a3

Problema 17

lim x 1 lim x 1

x 2 + 2x - 3 x 2 -1

(x + 3)(x - 1) x 2 + 2x - 3 = lim x 1 2 -1 (x - 1)(x + 1) xx2 - 1 = (x - 1) * (x + 1) Factorizacion dos nmeros sumados sea 2 y multiplicados sea -3 x2 + 2x - 3 = (x + 3) * (x - 1)


(x + 3) (x + 1)


=

1+ 3 4 = =2 1+1 2

9

lim x 1

x 2 + 2x - 3 =2 x 2 -1

Problema 18lim x a x3 - a3 x -a

lim x a

x3 - a3 = lim x a x -a

(x - a ) x 2 + ax + a 2

(x

- a)

Diferencia de cubos x3 a3 = (x - a) * (x2 + ax + a2) Factorizacion dos nmeros sumados sea 2 y multiplicados sea -3 x2 + 2x - 3 = (x + 3) * (x - 1)


lim x a = x 2 + ax + a 2Reemplazando el valor de (1)

= a 2 + a (a ) + a 2 = a 2 + a 2 + a 2 = 3 a 2x3 - a3 =3a2 x -a

lim x a

Problema 19 1 3 lim x 1 1- x 1- x3

Diferencia de cubos 1 x3 = (1 - x) * (1 + x + x2)

1 1 3 3 = lim x 1 lim x 1 1- x 1 - x (1 - x ) 1 + x + x 2 1- x3 1 + x + x 2 - 3 1+ x + x2 - 3 = lim x 1 lim x 1 (1 - x ) 1 + x + x 2 (1 - x ) 1 + x + x 2 (x + 2)(x - 1) x2 + x - 2 = lim x 1 lim x 1 (1 - x ) 1 + x + x 2 (1 - x ) 1 + x + x 2 (x + 2)(1 - x ) lim x 1 (1 - x ) 1 + x + x 2 Simplificando trminos semejantesFactorizacion dos nmeros sumados sea 1 y multiplicados sea -2

x2 + x 2 = (x + 2) (x 1)

10

lim x 1 lim x 1

(x + 2 ) 1 + x + x 2 x2 2 1+ x + x


1 2 = 1 + 1 + (1)2 lim x 1

-3 = = -1 3

1 3 = -1 1- x 3 1- x

Problema 20

lim x - 2

x 3 + 4x 2 + 4x (x + 2)(x - 3)x x 2 + 4x + 4 (x + 2)(x - 3)

x 3 + 4x 2 + 4x = lim x - 2 lim x - 2 (x + 2)(x - 3) lim x - 2 x (x + 2 )(x + 2 ) (x + 2)(x - 3)

x3 + 4x2 + 4x = (x ) * (x2 + 4x + 4) Factorizacion dos nmeros sumados sea 4 y multiplicados sea 4 x2 + 4x + 4 = (x + 2) * (x + 2) x3 + 4x2 + 4x = (x ) * (x + 2) * (x + 2)

Simplificando trminos semejanteslim x - 2 x (x + 2 ) (x - 3)

Reemplazando el valor de (1)= - 2 (- 2 + 2 ) - 2 (0 ) 2(0 ) 0 = = = =0 (- 2 - 3) -5 5 5

lim x - 2

x 3 + 4x 2 + 4x =0 (x + 2)(x - 3)

Problema 21lim x 1 x -1 x -1

conjugado

11

lim x 1

(

)( x + 1) (x - 1) ( x + 1)x -1+1

lim x 1

(x - 1) (x - 1) ( x

)


lim x 1

(=

1

x +1

)


=

( 1 + 1)

1

1 1 = 1+1 2x -1 1 = x -1 2

lim x 1

Problema 22lim x 4 x -2 4-x

Conjugado

lim x 4

(

)( x + 2) (4 - x ) ( x + 2)x -2+2

lim x 4

(x - 4) (4 - x ) ( x

)

= lim x 4

(4 - x ) (

- (4 - x ) x +2

)


lim x 4

( )

-1 x +2

)


=

(

1 4 +2

=

-1 -1 = 2+2 4

lim x 4

x -2 1 =4-x 4

12

Problema 23

lim

x 4

lim

x 4

x-4 x - x - 12 (x - 4 ) (x - 4) = lim x 4 2 (x - 4)(x + 3) x - x - 122

Factorizacion dos nmeros sumados sea -1 y multiplicados sea -12 x2 - x -12 = (x - 4) * (x + 3)


lim

x 4

1 ( x + 3)


=

1 1 = 4+3 7x 4

lim

x-4 1 = x 2 - x - 12 7

Problema 24

lim x 3 lim x 3

x 3 27 x2 - 9 = lim x 3 x 2 - 32 x 3 - 33

(x - 3) (x 2 + 3x + 9) (x 3)(x + 3)

Diferencia de cubos x3 33 = (x - 3 ) * (x2 + 3x +9) Diferencia de cuadrados x2 32 = (x - 3 ) * (x + 3)


lim x 3

(x 2 + 3x + 9)(x + 3 )


=

(3)2 + (3)(3) + 9 = 9 + 9 + 9 = 27 = 9 (3 + 3) 6 6 2x 3 27 x -92

lim x 3

=

9 2

Problema 25

lim h 0

(x + h )2 - x 2h

lim h 0

(x + h )2 - x 2 = limh

x 2 + 2xh + h 2 - x 2 x 0 h13


lim h 0

(h )(2x + h ) 2xh + h 2 = lim x 0 h h lim h 0 2x + h= 2x + 0 = 2xlim h 0


(x + h )2 - x 2h

= 2x

Problema 26

lim x 2

4 - x2 3 x2 + 5

Conjugado al denominador

3 - x 2 + 5 3 + 4 - x 2 3 + x 2 + 5 4 x 2 3 + = lim lim x 2 x 2 2 9- x +5 9 - x2 3 x +52

lim x 2

4 - x2

= lim x 2

(4 - x 2 ) 3 +( )

x2 + 5 x2 + 5

(

)

(

)

x2 + 5 4 - x2 = lim x 2 -5

(

) 3 + x 2 + 5 (4 x 2 )


lim x 2 3 + x 2 + 5Reemplazando el valor de (0)

= 3 + 22 + 5 = 3 + 4 + 5 = 3 + 9 = 3 + 3 = 6

lim x 2

4 - x2 3 x +52

=6

Problema 27

lim x 1lim x 1

x2 + x 2

(x - 1)2x2 + x - 2

(x - 1)

2

= lim x 1

(x + 2)(x - 1) (x 1)(x - 1)

Factorizacion dos nmeros sumados sea 1 y multiplicados sea 2 x2 + x -2 = (x +2) * (x -1)(x - 1)2 = (x - 1) * (x -1)


14

lim x 1

(x + 2 ) (x 1)


=

1+ 2 3 = = 11 0 x2 + x 2 = (No existe el lmite)

lim x 1

(x - 1)2

Problema 28

lim x 2 x 2 - 4xReemplazando el valor de (1)

= (2)2 - 4(x ) = 4 - 8 = - 4lim x 2 x 2 - 4x = - 4Problema 29

lim x - 1

(x 3 + 2x 2 3x - 4)

Reemplazando el valor de (-1)

= (- 1)3 + 2(- 1)2 - 3(- 1) - 4 = - 1 - 8 = - 1 + 2 + 3 - 4 = 0lim x - 1 x 3 + 2x 2 3x - 4 = 0Problema 30

(

)

lim x 1 lim x 1 lim x 1

(3x - 1)2 (x + 1)3 (3x - 1)2 (x + 1)3

= lim x 1

(3x - 1)(3x - 1) (x + 1)39x 2 - 6x + 1

9x 2 3x - 3x + 1

(x + 1)3

= lim x 1

(x + 1)3

Formando un cuadrado perfecto (sptimo caso de factorizacion)

se multiplica toda la expresin por 9 y luego se divide por 9

9x 2 - 6x + 1 =

9 9x 2 - 9(6x ) + 9(1) (9x )2 - 6(9x ) + 9 = 9 915

( )

Dos nmeros que sumados sean -6 y multiplicados sea 9 (9x)2 6x + 9 = (9x - 3 ) (9x - 3 )

9 x 2 - 6x + 1 =

(9x - 3) (9x - 3) = (9x - 3)(9x - 3) = 3(3x - 1) * 3(3x - 1)9 3*3 3*3


9x 2 - 6x + 1 = (3x - 1) * (3x - 1) Reemplazando (3x 1) * (3x - 1) 9x 2 - 6x + 1 lim x 1 = lim x 1 3 (x + 1) (x + 1)3Reemplazando el valor de (1)

=

[3(1) 1]* [3(1) - 1] = [2]* [2] = 4 = 1 (1 + 1)3 (2)3 8 2(3x - 1)2 (x + 1)3= 1 2

lim x 1

Problema 31

lim x 0

3 x - 3- x

lim x 0

(3x )(3x )- 1 3 x = lim 3x x 0 x x 1 (3 )(3 )+ 1 3x + x3x 1 3 3x

3 x + 3- x

Diferencia de cubos x3 a3 = (x - a) * (x2 + ax + a2)


Factorizacion dos nmeros sumados sea 2 y multiplicados sea -3 x2 + 2x - 3 = (x + 3) * (x - 1)

lim x 0

3 2x - 1 32x + 1


=

1 -1 0 = =0 3 2(0 ) + 1 30 + 1 1 + 1 2 = = 3 x - 3- x 3 x + 3- x =0

3 2(0 ) - 1

30 - 1

lim x 0

16

Problema 32

lim x 2

x -1 x 2 -1 x -1 x -12

lim x 2

= lim x 2

x 1 (x - 1)(x + 1)Diferencia de cuadrados x2 12 = (x - 1 ) * (x + 1)


lim x 2

1 (x + 1)


=

1 1 = 2 +1 3

lim x 2

x -1 x 2 -1

=

1 3

Problema 33

lim x 0Conjugado

1+ x - 1- x x

lim x 0

( 1+ x(x ) (

(x ) (

- 1- x

)( 1 + x)

+ 1- x

1+ x + 1- x

)

) = lim x 02x

(1 + x ) - (1 - x ) (x ) ( 1 + x + 1 - x )

lim x 0

1 -1 + x + x 1+ x + 1- x

= lim x 0

x

( 1+ x +

1 x

)


lim x 0

( 1+ x +2 + 1 0

2

1 x

)2 =


=

( 1+ 0

)

=

2 2 = =1 1 + 1 1+1 2

lim x 0

1+ x - 1- x =1 x

17

Problema 34

lim x 7Conjugado

2- x -3 x 2 - 49

lim x 7

(2 -

)( (x 2 - 7 2 )(2 +4-x +3 -72

x -3 2+ x -3 x -3

)

) = lim x 7

(x 2 - 7 2 )(2 +7x

4 - (x - 3) x -3

)= lim x 7

lim x 7

(x

2

)(2 +

x -3

)

= lim x 7

(x - 7 )(x + 7 ) (2 +

x -3

)

(x - 7 )(x + 7 ) (2 +

(x 7 )

x -3

)


lim x 7

(x + 7 ) (2 +1

1

x -3

)-1


=

(7 + 7 ) (2 +

7-3

)

=

(14) * (2 +

4

)

=

-1 -1 = 14 (4) 56

lim x 7

2- x -3 x 2 - 49

=

-1 56

Problema 35

x -1 lim x 1 3 x -1

Se utiliza un cambio de variable x = t6 x 1 t 1

x -1 t 6 -1 t 3 -1 lim x 1 = lim t 1 = lim t 1 3 x -1 3 6 t 2 -1 t -1lim x 1

(t - 1) (t + t 2 + 1) (t - 1)(t + 1)

Diferencia de cubos t3 13 = (t - 1) * (t + t2 + 1) Diferencia de cuadrados t2 12 = (t - 1 ) * (t + 1)


lim x 1

(t + t 2 + 1)(t + 1)


=

1 + (1)2 + 1 3 = 1+1 218

x -1 3 = lim x 1 3 x -1 2Problema 36

lim x 4Conjugado

3- 5+ x 1- 5 - x

lim x 4

(3 (1 -

5 + x 3 + 5 + x 1+ 5 - x 9 - (5 + x ) 1 + 5 - x = lim x 4 [1 - (5 - x )] 3 + 5 + x 5 - x 3 + 5 + x 1+ 5 - x

)( )(

)( )(

) )

[

(

(

)] )

lim x 4

(9 - 5 - x ) (1 + (1 - 5 + x ) (3 + (x - 4) (3 +

) = lim x 4 (4 - x ) (1 + (x - 4) (3 + 5+ x)5-x

) 5+ x)5-x

lim x 4

(x - 4 ) 1 + 5 - x

(

) 5+ x)


lim x 4

1+ 5 - x

(3 +

(

) 5+ x)


=

1+ 5 - 4

(3 +

(

) = - (1 + 1) = - (2) = - 2 = - 1 (3 + 9 ) 3 + 3 6 3 5 + 4)3- 5+ x 1- 5 - x = -1 3

lim x 4

Problema 37

lim x 0Conjugado

1 + x + x 2 + x -1 x 2 1 + x + x 2 + (x - 1) 1 + x + x + (x - 1) = lim x 0 x x 1 + x + x 2 1 + x + x 2 - (x - 1) - (x - 1)

lim x 0

19

1 + x + x 2 - (x - 1)2 lim x 0 = lim x 0 2 - (x - 1) x 1 + x + x Simplificando trminos semejantes

1 + x + x 2 - x 2 2 x + 1 = lim x 0 2 - (x - 1) x 1 + x + x

1 + x + x 2 - x 2 + 2x - 1 x 1 + x + x 2 - (x - 1)

lim x 0

3x x 1 + x + x 2 - (x - 1) 3 2 1 + x + x - (x - 1)

lim x 0


=

3 1 + 0 + (0)2 - (0 - 1)

=

( 1) + 1

3

=

3 2

lim x 0

1 + x + x 2 + x -1 3 = x 2

Problema 38

lim x 0Conjugado

x x + 1 -1

lim x 0

(x

x

(

x +1 +1

x +1 -1

)()

)

x +1 +1

)

= lim x 0

x x +1 +1 (x + 1) - 1

(

)

lim x 0

(

x +1 +1 x = lim x 0 x + 1-1

(

x +1 +1 x

)


lim x 0

x +1 +1


=

0 +1 +1=

1 +1= 2x x + 1 -1 =2

lim x 0

20

Problema 39 x+2- 2 lim x 0 x+5 5 Conjugadolim x 0

( x + 2 2 )( x + 2 + 2 )( x + 5 + 5 ) = lim x 0 [(x + 2 - 2) ( x + 5 + 5 )] ( x + 5 5 )( x + 2 + 2 )( x + 5 + 5 ) [(x + 5 - 5)] ( x + 2 + 2 )x


( x + 5 + 5 ) = lim x 0 ( x + 5 + 5 ) ( x + 2 + 2) x ( x + 2 + 2)5 2 = 5 2

Reemplazando el valor de (0)=

( 0 + 5 + 5 ) = - (2 5 ) = ( 0+ 2 + 2) (2 2)x+2 - 2 x +5 5 = 5 2

lim x 0

Problema 40lim x 0 4+ x -2 x

Conjugado

lim x 0

((

4+x -2 = lim x 0 x

)

[(

4+x -2

x 4+x +2

[

)][()

4+x + 2

]

)]

lim x 0

(4 + x - 4)

x 4+x +2

)

= lim x 0

x 4+x +2

(

x


lim x 0

(

1 4+x +2

)


=

(

1 4+0 +2

) (

=

1 4 +2

)

=

1 4

21

lim x 0

4+ x -2 1 = x 4

Problema 41lim x 3 x -3 3 - 18 - x 2

Conjugado 3 - 18 - x 2 3 + 18 - x 2

(x 3) 3 +

lim x 3

18 - x 2

(x - 3) 3 + = lim x 3

18 - x 2 9 - 18 - x 2

(x - 3) 3 + lim x 3

(x - 3) 3 + 18 - x 2 18 - x 2 = lim x 3 2 -9 2 x 9 - 18 + x

(x - 3) 3 + lim x 3

18 - x 2 (x - 3)(x + 3)

Diferencia de cuadrados x2 32 = (x - 3 ) * (x + 3)


3 + 18 - x 2 lim x 3 (x + 3)Reemplazando el valor de (3)3 + 18 - (3)2 3+3 3 + 18 - 9 3 + 9 3 + 3 6 = = = =1 6 6 6 6 =1

=

=

lim x 3

x -3 3 - 18 - x 2

Problema 42 x-a lim x a a x - a Reemplazando el valor de (a)

=

aa a a- a

=

0 a a a

=0

22

lim x a

x-a a x - a

=0

23

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problemas-resueltos-limites