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Ejercicio 1.3 En los ejercicios 1 al 4, utilizar una herramienta de graficación para representar la función y estimar los límites de manera visual. 1. h(x) = -x 2 + 4x a. () b. () = -(4) 2 + 4(4) = -(-1)2 + 4(-1) = - 16 + 16 = - (1) - 4 = 0 = -5 2. g(x) = ( √ ) a. () b. () ( √ ) ( √ ) = 12 = 12 = 12 ( ) = 12 ( ) = 12 ( ) = 12 ( ) = 12 ( ) = 12 ( ) = 12 ( ) = 12 ( ) = = = 4

Seccion 1.3 Primera Parte Larson Calculo 1

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Page 1: Seccion 1.3 Primera Parte Larson Calculo 1

Ejercicio 1.3

En los ejercicios 1 al 4, utilizar una herramienta de graficación para representar la función

y estimar los límites de manera visual.

1. h(x) = -x2 + 4x

a. ( ) b. ( )

= -(4)2 + 4(4) = -(-1)2 + 4(-1)

= - 16 + 16 = - (1) - 4

= 0 = -5

2. g(x) = ( √ )

a.

( ) b.

( )

( √ )

( √ )

= 12

= 12

= 12 (√

) = 12 (

)

= 12 (

) = 12 (

)

= 12 (

) = 12 (

)

= 12 (

) = 12 (

)

=

=

= 4

Page 2: Seccion 1.3 Primera Parte Larson Calculo 1

3. f(x) = x Cos x

a.

( ) b.

( )

= 0 (Cos 0) = π 3 (Cos π 3)

= 0 (1) ≈ 0,52

= 0

4. f(t) = t | t – 4 |

a.

f(t) b.

f(t)

= 4 | 4 – 4 | = -1 -1 -4

= 4 | 0 | = -1 5

Page 3: Seccion 1.3 Primera Parte Larson Calculo 1

= 0 = -1 (5)

= -5

En los ejercicios 5 al 22, calcular el límite

5.

x3 6.

x4

= (2) 3 = (-2)4

= 8 = 16

7.

(2x – 1) 8.

(3x + 2)

= 2 (0) – 1 = 3(-3) + 2

= 0 – 1 = - 9 + 2

= -1 = - 7

9.

( 3 ) 10. (

1)

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

= 9 – 6 = - 1 + 1

= 3 = 0

11.

(2 + 4x + 1) 12.

(3 - 2 + 4)

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

= 2 (9) – 12 + 1 = 3(1) – 2(1) + 4

= 18 – 1x = 3 – 2 + 4

= 7 = 5

13.

√ 1 14.

= √3 1 = √

Page 4: Seccion 1.3 Primera Parte Larson Calculo 1

= √ = √

= 2 = 2

15.

( 3) 16.

(2 1)

= (3 + 3)2 = (2(0) – 1)3 = (6)2 = (0 – 1)3

= 36 = (-1)3

= -1

17.

18.

=

=

=

= - 2

19.

20.

=

=

( )

=

=

= -

21.

√ 22.

= ( )

√ =

=

√ =

=

=

= 7 = - 1

En los ejercicios 23 al 26, encontrar los límites

23. f(x) = 5 – x, g(x) =

a.

f(x) b.

( ) c.

( ( ))

5 – x

g (5-x)

= 5 -1 = ( ) = (5 – x)3

= 4 = 64 = (5 – 1)3

= (4)3

= 64

Page 5: Seccion 1.3 Primera Parte Larson Calculo 1

24. f(x) = x + 7, g(x) =

a.

f(x) b.

( ) c.

( ( ))

x + 7

g (x + 7)

= -3 + 7 = ( )

( )

= 4 = 16 = (-3 + 7)2

= (4)2

= 16

25. f(x) = 4 - , g(x) = √ 1

a.

f(x) b.

g(x) c.

( ( ))

4 -

√ 1

g (4 - )

= 4 - (1) = √3 1

√ 1

= 4 – 1 = √ = √5 (1)

= 3 = 2 = √5 1

= √

= 2

26. f(x) = 2 3 1, g(x) = √

a.

f(x) b.

g(x) c.

( ( ))

2 3 1

(2 3 1)

= 2( ) – 3(4) + 1 = √21

= √2 3 1

= 2(16) – 12 + 1 = √2

= √2( ) 3( )

= 32 – 11 = 3 = √2(1 ) 12

= 21 = √32 5

= √2

= 3

En los ejercicios 27 a 36, encontrar el límite de la función trigonométrica

27.

Sen x 28.

Tan x

= Sen

= Tan π

= 1 = -1

29.

Cos

30.

Sen

= Cos ( )

= Sen

( )

Page 6: Seccion 1.3 Primera Parte Larson Calculo 1

= Cos

= Sen

=

= 0

31.

Sec 2x 32.

Cos 3x

=

= Cos 3π

=

( ) = -1

=

=

= 1

33.

Sen x 34.

Cos x

= Sen

= Cos

=

=

35.

Tan (

) 36.

Sec (

)

=

=

=

=

,

= - 1 ≈ -1,15

En los ejercicios 37 a 40, utilizar la información que se expone para evaluar los límites

37.

f(x) = 3,

g(x) = 2

a.

[ 5 g(x) ] b.

[f(x) + g(x)] c.

[f(x) . g(x)] d.

( )

( )

[ 5(2) ]

[3 + 2]

3 · 2 ] =

= 10 = 5 = 6

38.

f(x) =

,

g(x) =

a.

[4 f(x)] b.

[f(x) + g(x)] c.

[f(x) . g(x)] d.

( )

( )

Page 7: Seccion 1.3 Primera Parte Larson Calculo 1

4 (

)]

[

]

[

]

= 6 =

=

=

= 2 = 3

39.

f(x) = 4

a.

[ f(x)]3 b.

√ ( ) c.

[3 f(x)] d.

( ) ⁄

[ 4 ]3 b.

√ c.

[3 (4)] d.

= 64 = 2 = 12 = √

= 8

40.

f(x) = 27

a.

√ ( ) b.

( )

c.

( ) d.

( ) ⁄

√2

[ 27 ]2

√2

= 3 =

= 729 = √ 2

= 9

En los ejercicios 41 a 44 utilizar la gráfica para determinar el límite (si existe) de manera

visual. Escribir una función más simple que coincida con la dada, salvo en un punto.

41. g(x) =

a.

g(x) b.

g(x)

= ( )

=

( ) ( )

= x – 1 =

= 0 – 1 =

= -1 = -2

g(x) =

y f(x) = x – 1, no coinciden en x = 0

42. h(x) =

a.

h(x) b.

h(x)

Page 8: Seccion 1.3 Primera Parte Larson Calculo 1

= ( ) ( )

=

( )

=

= - x + 3

=

= 3

= 1

g(x) =

y f(x) = - x + 3, no coinciden en x = 0

43. g(x) =

a.

g(x) b.

g(x)

( )

=

( ) ( )

( )( )

=

x (x + 1) =

= 1 (2) = 0

= 2

g(x) =

y f(x) = + x, coinciden excepto en x = 1

44. f(x) =

a.

f(x) b.

f(x)

( )

( )

=

=

=

= - 1

= ∞

El

no existe

Page 9: Seccion 1.3 Primera Parte Larson Calculo 1

En los ejercicios 45 a 48, encontrar el límite de la función (si existe). Escribir una función

más simple que coincida con la dada salvo en un punto. Utilizar una herramienta de

graficación para verificar el resultado.

45.

( )( )

x – 1

= - 1 – 1

= - 2

f(x) =

y g(x) = x – 1, coinciden excepto en x = -1

46.

(2 3) ( 1)

1

2x – 3

= 2( -1) – 3

= - 2 – 3

= -5

f(x) =

y g(x) = 2x – 3, coinciden excepto en x = -1

47.

( )( )

2

= (2) 2(2)

= 4 + 4 + 4

= 12

F(x) =

y g(x) = 2 , coinciden excepto en x = 2

Page 10: Seccion 1.3 Primera Parte Larson Calculo 1

48.

( )( )

1

= ( 1) ( 1) 1

= 1 + 1 + 1

= 3

f(x) =

y g(x) = 1, coinciden excepto en x = -1

En los ejercicios 49 a 64, encontrar el límite (si existe)

49.

50.

( )

( )

=

=

= - 1

51.

52.

( )( )

( )

( )( )

=

=

=

=

53.

54.

( )( )

( )( )

( )( )

( )( )

=

=

=

=

=

=

Page 11: Seccion 1.3 Primera Parte Larson Calculo 1

55.

56.

·

·

( √ ) ( )

(√ )

( √ ) ( )

(√ )

(√ )

(√ )

(√ )

(√ )

=

√ =

=

√ =

=

=

=

=

57.

√ √

58.

√ √

√ √

·

√ √

√ √

√ √

·

√ √

√ √

( √ ) (√ )

(√ √ )

( √ ) (√ )

(√ √ )

(√ √ )

(√ √ )

(√ √ )

(√ √ )

√ √

√ √

=

√ √

√ √

=

√ ·

√ =

√ ·

= √

√ =

= √

=

59.

( )

60.

( )

( )

( )( )

( )

( )( )

Page 12: Seccion 1.3 Primera Parte Larson Calculo 1

·

·

=

( ) =

( )

=

=

61.

( )

62.

( )

( )

·

( )

( )

( ) ( )

( )

( )

( )

2x +

= 2x + 0

( )

= 2x

= ( )

( )

=

=

= 2x

63.

( ) ( ) ( )

64.

( )

( )

( )

3 3

= 2x + 0 – 2 = 3 + 3x(0) + (0) 2

= 2x – 2 = 3

En los ejercicios 65 a 76, determinar el límite (si existe) de la función trigonométrica.

65.

66.

( )

·

= 3

Page 13: Seccion 1.3 Primera Parte Larson Calculo 1

= (

)

= 3 (0)

= (

) (1) = 0

=

67.

( )

68.

·

·

= (1) · 0 = 1

= 0

69.

70.

· Senx

= 1 · Sen 0

·

= 1 (0)

( )

= 0

·

= (1) ·

= (1) ·

= 0

71.

( )

72.

· 1 – Cos h

·

·

1 – Cos h = ·

= (0) · 1 – Cos 0 = ·

= 0 · 1 – 1 = -

= 0

73.

74.

Page 14: Seccion 1.3 Primera Parte Larson Calculo 1

Cos x ·

(

)

= Sen 2

( )

·

= 1

-

=

= √2