seccion 1.3 primera parte larson calculo 1
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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) = -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
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
= 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 = √
= √ = √
= 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
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
( )
= 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.
( )
( )
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)
= ( ) ( )
=
( )
=
= - 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
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
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.
( )( )
( )( )
( )( )
( )( )
=
=
=
=
=
=
55.
√
56.
√
√
·
√
√
√
·
√
√
( √ ) ( )
(√ )
( √ ) ( )
(√ )
(√ )
(√ )
(√ )
(√ )
√
√
=
√ =
√
=
√ =
√
=
=
=
=
57.
√ √
58.
√ √
√ √
·
√ √
√ √
√ √
·
√ √
√ √
( √ ) (√ )
(√ √ )
( √ ) (√ )
(√ √ )
(√ √ )
(√ √ )
(√ √ )
(√ √ )
√ √
√ √
=
√ √
√ √
=
√ ·
√
√ =
√ ·
√
√
= √
√ =
√
√
= √
=
√
59.
( )
60.
( )
( )
( )( )
( )
( )( )
·
·
=
( ) =
( )
=
=
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
= (
)
= 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.
Cos x ·
(
)
= Sen 2
( )
·
= 1
-
=
√
= √2
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