lista_calculo1_02

Prévia do material em texto

Ca´lculo I
Unifesp - 1o semestre de 2012
Lista de Exerc´ıcios 2
1. Usando as propriedades de limites determine os seguintes valores:
(a) lim
x→7
(2x+ 5)
(b) lim
t→6
8(t− 5)(t− 7)
(c) lim
h→0
1√
x+ h+
√
x
(d) lim
y→−5
y2
5− y
(e) lim
h→0
3√
3h+ 1 + 1
(f) lim
x→2
x− 2
x2 − 4
(g) lim
x→2
x2 − 3x+ 2
x− 2
(h) lim
y→1
5y3 + 8y3
3y4 − 16y2
(i) lim
x→−4
(
2x
x+ 4
+
8
x+ 4
)
(j) lim
y→0
5y3 + 8y2
3y4 − 16y2
(k) lim
t→0
t+ a/t
t+ b/t
(l) lim
x→4
4x− x2
2−√x
(m) lim
x→1
√
x2 + 1−√2
x− 1
(n) lim
x→−2
x+ 2√
x2 + 5− 3
(o) lim
x→1
x2 − 1
x2 − 2x+ 1
(p) lim
x→−4
(
2x
x+ 4
− 8
x+ 4
)
(q) lim
x→−4
sen
(
x2 − 16
x+ 4
pi
)
2. Se
√
5− 2x2 ≤ f(x) ≤ √5 + 2x2 para −1 ≤ x ≤ 1, determine lim
x→0
f(x), por mais
complicada que possa ser f(x). (Sugesta˜o: use o teorema do confronto).
3. Seja a func¸a˜o
h(x) =

x, se x < 1
3, se x = 1
2− x2, se 1 < x ≤ 2
x− 3, se x > 2
Calcule
(a) lim
x→1−
h(x)
(b) lim
x→1
h(x)
1
(c) h(1)
(d) lim
x→2−
h(x)
(e) lim
x→2+
h(x)
(f) lim
x→2
h(x)
4. Usando as propriedades dos limites, determine o valor dos seguintes limites laterais.
(a) lim
x→− 1
2
−
√
x+ 2
x+ 1
(b) lim
x→1−
(
1
x+ 1
)(
x+ 6
x
)(
3− x
7
)
(c) lim
h→0+−
√
h2 + 4h+ 5−√5
h
(d) lim
h→0+−
√
6−√5h2 + 11h+ 6
h
(e) lim
x→−2−
(x+ 3)
|x+ 2|
x+ 2
(f) lim
x→−2+
(x+ 3)
|x+ 2|
x+ 2
(g) lim
x→1+
√
2x(x− 1)
|x− 1|
(h) lim
x→1−
√
2x(x− 1)
|x− 1|
5. Calcule o valor dos seguintes limites envolvendo infinitos:
(a) lim
x→∞
pi − 2
x2
(b) lim
x→∞
2− 2/x
5 +
√
3/x2
(c) lim
r→∞
r + sen(r)
2r + 7− 5 sen(r)
(d) lim
x→−∞
exsen
(
1
x
)
(e) lim
x→−∞
9x4 + x
2x4 + 5x2 − x+ 6
(f) lim
x→∞
x2 − 5x+ 1
3x2 + 7
(g) lim
x→∞
2x+ 3
x+
√
x
(h) lim
x→∞
2x5/3 − x1/3 + 7
x8/5 + 3x+
√
x
(i) lim
x→+∞
x+ 2√
4x2 + x− 2
(j) lim
x→−∞
x+ 2√
4x2 + x− 2
6. Determine os limites infinitos abaixo e a equac¸a˜o da reta ass´ıntota vertical em cada
caso.
(a) lim
x→0
1
3x
(b) lim
x→7
4
(x− 7)2
(c) lim
x→0
−1
x2(x+ 1)
(d) lim
x→0−
2
x
1
5
(e) lim
x→(pi2 )
−
[tg(x)]
(f) lim
x→0−
(1 + cossec(x))
2
7. Usando os limites fundamentais
lim
x→0
sen(x)
x
= 1 e lim
x→∞
(
1 +
1
x
)x
= e
calcule
(a) lim
x→0
sen(3x)
4x
(b) lim
x→0
tg(2x)
x
(c) lim
x→0−
x+ x cos(x)
sen(x)cos(x)
(d) lim
x→0
sen(x)
sen(2x)
(e) lim
x→0
sen(sen(x))
sen(x)
(f) lim
x→0
sen(3x)cotg(5x)
x2
(g) lim
n→∞
(
1 +
2
n
)n
(h) lim
n→∞
(
1 +
1
n
)n+3
(i) lim
n→∞
(
1 +
3
n
)n+2
(j) lim
n→∞
(
n+ 7
n+ 4
)n
8. Em quais intervalos as func¸o˜es abaixo sa˜o cont´ınuas?
(a) f(x) = 1
x−2 − 3x
(b) f(x) = x+4
x2−3x−10
(c) f(x) = |x− 1|+ sen(x)
(d) f(x) = 2+x
cos(x)
(e) f(x) =
√
2x+ 3
(f) f(x) = x tg(x)
x2+1
9. Dada a curva y = x+1
x−5 , determine:
(a) limites laterais para x→ 5,
(b) limites no infinito,
(c) ass´ıntotas horizontal e vertical,
(d) esboc¸o do gra´fico.
10. Em um reator qu´ımico, a concentrac¸a˜o de uma substaˆncia varia no tempo de acordo
com a expressa˜o:
C(t) =
50t
200 + t
,
onde C representa a concentrac¸a˜o em mg/m3 e t representa o tempo em minutos.
Apo´s um tempo suficientemente longo, verificou-se que a concentrac¸a˜o da substaˆncia
se estabilizou. Em que valor a concentrac¸a˜o se estabilizou?
11. Calcule
lim
x→c
f(x)− f(c)
x− c
se
3
(a) f(x) = 2x2 − 3x, c = 2. Resposta 5
(b) f(x) =
√
x, c = 4. Resposta 1
4
Respostas
1. (a) 19
(b) −8
(c) 1
2
√
x
(d) 5
2
(e) 3
2
(f) 1
4
(g) 1
(h) −1
(i) 2
(j) −1
2
(k) a
b
(l) 16
(m) 1√
2
(n) −3
2
(o) Na˜o existe
(p) Na˜o existe
(q) 0
(r) 1
2
(s) −1
2
2.
√
5
3. (a) 1
(b) 1
(c) 3
(d) −2
(e) −1
(f) na˜o existe
4. (a)
√
3
(b) 1
(c) 2√
5
(d) − 5√
6
(e) −1
(f) 1
(g)
√
2
(h) −√2
5. (a) pi
(b) 2
5
(c) 1
2
(d) 0
(e) 9
2
(f) 1
3
(g) 2
(h) ∞
6. (a) ∞
(b) ∞
(c) −∞
(d) −∞
(e) ∞
(f) −∞
7.
4
(a) 3
4
(b) 2
(c) 2
(d) 1
2
(e) 1
(f) 12
(g) e2
(h) e
(i) e3
(j) e3
8. (a) S = {x ∈ R|x 6= 2}
(b) S = {x ∈ R|x 6= 5 ou x 6= −2}
(c) S = {x ∈ R}
(d) S = {x ∈ R|x 6= (2n+ 1)pi
2
,∀n ∈ Z}
(e) S = {x ∈ R|x ≥ −3
2
}
(f) S = {x ∈ R|x 6= (2n+ 1)pi
2
,∀n ∈ Z}
9. (a) +∞ (x→ 5+);−∞(x→ 5−)
(b) 1
(c) Ass´ıntota vertical: x = 5; Ass´ıntota
horizontal: y = 1
10. (a) 50mg/m3
5