Practica Calculo I

´
Instituto Tecnologico de Costa Rica
´
Escuela de Matematica

´
Calculo Diferencial e Integral
II Semestre de 2004

Ejercicios sobre l´
ımites
Determinaci´n de l´
o
ımites de una funci´n dada su gr´fica
o
a
Considere las funciones siguientes y sus representaciones gr´ficas. En cada caso, y si existen, determine a
a
partir de la gr´fica los l´
a
ımites que se indican.
(a)

y3

lim f (x)

x→−3+

(b) lim f (x)
x→−1

2

1.

(c) lim f (x)
x→2

1
x
-3

-2

-1

1

2

3

4

5

(e)

(a)

3

(b)

2
1

1

-1,5

2.

(d) f (−1); f (2)
lim f (x)

x→+∞

lim f (x)

x→−∞

lim f (x)

x→−3/2

(c) lim f (x)
x→3/2

1,5
-1

(d) f (3/2)
(e)

(a)

2

lim f (x)

x→+∞

lim f (x)

x→−∞

(b) lim f (x)
x→−21

(c) lim f (x)
-2

-1

1

x→−1

3

2

(d) lim f (x)

3.

x→0

-1

(e) lim f (x)
x→2

(f) lim f (x)
x→3

(g)

1

lim f (x)

x→+∞

(a)

(b) lim f (x)

2,5

x→−2

2
-2

4.

lim f (x)

x→−∞

(c) lim f (x)

-1

x→−1

1

(d) lim f (x)
x→0

(e) lim f (x)

-2

x→1

(f)

(a)

lim f (x)

x→+∞

lim g (x)

x→−∞

(b) lim g(x)
x→−3

3

(c) lim g (x)
x→−1

1

5.

-1

-2

1

(d) lim g (x)

2

-3

x→0

4

(e) lim g (x)
x→1

-2

(f) lim g (x)
x→2

(g)

(a)
2

lim g (x)

x→+∞

lim h(x)

x→−∞

(b) lim h(x)
x→−3

6.

-3

(c) lim h(x)
x→−2

-2

(d) lim h(x)

-2

x→0

(e)

2

lim h(x)

x→+∞

(a)

lim f (x)

x→−∞

(b) lim f (x)
x→−2

2

(c)lim f (x)

7.

x→0

1

(d) lim f (x)
x→2

-2

2

(e)

lim f (x)

x→+∞

Construcci´n de la gr´fica de una funci´n conociendo sus l´
o
a
o
ımites
En cada caso siguiente considere los datos indicados sobre la funci´n f y dibuje una gr´fica que la represente.
o
a
1.

• Dh = I − {−2, 2}
R

• f (−3) = 1
• lim h(x) = −∞

• lim h(x) = −∞

• lim h(x) = −2

•

•

•Df = I − 0
R
• lim f (x) = +∞

• lim f (x) = 0
• f (x) = 1, ∀x ∈]0, 1[

• f (2) = 2; f (3) = 1
• lim f (x) = +∞

• Dg =] − ∞, 1[∪]2, +∞[
• lim g (x) = 4

•

• lim g (x) = +∞

•

lim h(x) = −∞

x→−∞
x→−3

2.

x→−∞

3.

x→−2−

lim h(x) = +∞

x→−2+

x→0−

lim g (x) = +∞

x→−1+

x→2−

• lim h(x) = +∞
x→2+

lim h(x) = +∞

x→+∞

x→+∞

x→2+

• lim g(x) = +∞

•

• Df = I −] − 2, 2[
R
• lim f (x) = +∞

• lim f (x) = −3
• f (−4) = −2

• f (−2) = f (2) = 0
• lim f (x) = −2

• Dg = I − [0, 1]
R

• lim g (x) = −∞

• f (−3) = f (2) = 0

•

• lim g (x) = −∞

• f (3) = 3; f (4) = 5; f (5) =
4

•

• lim h(x) = +∞

x→−∞

•
4.

6.

lim g (x) = 3

x→+∞

• Dh =] − 3, +∞[−{−2, 3}
• lim h(x) = 0
x→3

•
7.lim h(x) = 4

x→−4

x→1+

x→0−

lim h(x) = 5

x→−2+

• Df = I
R
• lim f (x) = −2
• lim f (x) = 2
• Df = I − {0}
R
• lim f (x) = +∞
x→−∞

•

x→2+

• lim h(x) = −∞

•

• lim f (x) = −∞

• lim f (x) = 2

• lim f (x) = 4

• f (3) = 3 y f (−2) = 1.

x→2−

x →0

x→3+

lim h(x) = 2

x→+∞

lim f (x) = +∞

x→−2−

lim f (x) = −∞

x→−2+

x→3−

•limitex+∞f (x) = −2

x→−2

•

x→+∞

x→−2−

x→−∞

8.

lim g (x) = 5

x→+∞

x→−2−

x→−∞

5.

x→1−

lim g (x) = −∞

• lim f (x) = 1

• lim f (x) = 1

• lim f (x) = −1

• f (2) = 1
• lim f (x) = 3

x→0−
x→0+

• lim f (x) = 0
x→2−

3

x→2+

x→+∞

Calcule los siguientes l´
ımites (si existen). En caso de que no existan, justifique su respuesta.
√
3
5y − 15x3 + 2x2 − 5x − 6
1 − sen r − 1
√
20. lim
1. lim 3
39. lim
2 − 4x − 4
y →3 1 − 9 2y − 5
x→2 x + x
r→0
r
√
sen z
x2 + x − 6
2 − 6 3a + 64
40. lim
2. lim 3
21. lim
z →0 z + sen z
x→−3 x + 2x2 − 3x
a→0
5a
t − sen (2t)
√
41. lim
a3 − b − ab + a2
3 − 4 k + 82
t→0 t + sen (3t)
3. lim
22. lim √
b→a2 2a3 − 2ab + b − a2
k→−1 3 k + 28 − 3
1 − cos3 n
42. lim
2w3 − 4aw2 +...