lista limites c

Prévia do material em texto

Cálculo I - Prof
a
Marina Ribeiro
Lista 4 - LIMITES - PARTE II
1. Para cada uma das funções abaixo, calcule o limite quando: (i) x→ +∞ e (ii) x→ −∞
(a) f(x) =
2
x
− 3
(b) g(x) =
1
2 + 1/x
(c) h(x) =
−5 + 7/x
3− (1/x2)
(d) f(x) =
2x+ 3
5x+ 7
(e) f(x) =
x+ 1
x2 + 3
(f) f(x) =
7x3
x3 − 3x2 + 6x
(g) h(x) =
10x5 + x4 + 31
x6
(h) g(x) =
−2x3 − 2x+ 3
3x3 + 3x2 − 5x
(i) f(x) =
3x− |x|
7x− 5|x|
2. Calcule os limites abaixo:
(a) lim
x→+∞ 3x
3 + 4x2 − 1
(b) lim
t→+∞
t2 − 2t+ 3
2t2 + 5t− 3
(c) lim
x→−∞
3x5 − x2 + 7
2− x2
(d) lim
v→+∞
v
√
v − 1
3v − 1
(e) lim
x→+∞x(
√
x2 − 1− x
(f) lim
s→+∞
3
√
3s7 − 4s5
2s7 + 1
(g) lim
x→3+
x
x+ 3
(h) lim
y→6+
y + 6
y2 − 36
(i) lim
x→3−
1
|x− 3|
(j) lim
x→+∞
2
√
x+ x−1
3x− 7
(k) lim
x→−∞
2x5/3 − x1/3 + 7
x8/5 + 3x+
√
x
(l) lim
u→+∞
4u4 + 5
(u2 − 2)(2u2 − 1)
(m) lim
x→+∞
√
9x6 − x
x3 + 1
(n) lim
x→−∞x
4 + x5
(o) lim
x→+∞
1− ex
1 + 2ex
3. Esboce o gráfico de um função y = f(x) que satisfaça as condições dadas:
(a) f(0) = 0, f(1) = 2, f(−1), lim
x→−∞ f(x) = −1, limx→+∞ f(x) = 1
(b) f(0) = 0, lim
x→−∞ f(x) = 0, limx→+∞ f(x) = 0, limx→0+
f(x) = 2 e lim
x→0−
f(x) = −2
(c) f(0) = 0, lim
x→±∞ f(x) = 0, limx→1−
f(x) = lim
x→−1+
f(x) = +∞, lim
x→1+
f(x) = −∞, lim
x→−1−
f(x) = −∞
(d) f(2) = 1, f(−1) = 0, lim
x→−∞ f(x) = 0, limx→0+
f(x) = +∞, lim
x→0−
f(x) = −∞ e lim
x→+∞ f(x) = 1
4. Defina uma função que satisfaça as condições dadas:
(a) lim
x→−∞ f(x) = 0, limx→+∞ f(x) = 0, limx→2−
f(x) = +∞, lim
x→2+
f(x) = +∞
(b) lim
x→−∞ g(x) = 0, limx→+∞ g(x) = 0, limx→3−
g(x) = −∞ e lim
x→3+
g(x) = +∞
(c) lim
x→−∞ f(x) = −1, limx→+∞ f(x) = 1, limx→0− g(x) = −1 e limx→0+ f(x) = 1
(d) lim
x→−∞ f(x) = 1, limx→+∞ f(x) = 1, limx→1−
f(x) = +∞ e lim
x→1+
f(x) = −∞
(e) lim
x→−∞ g(x) = 0, limx→+∞ g(x) = 0, limx→0
g(x) = −∞, f(2) = 0, lim
x→3+
g(x) = −∞ e lim
x→3−
g(x) = +∞
(f) A função f possua por assíntotas verticais x = 1 e x = 3 e por assíntota horizontal y = 1.
5. Determine, caso exista, assíntotas horizontais e verticais de cada curva:
1
(a) y =
x
x+ 4
(b) y =
2x2 + x− 1
x2 + x− 2
(c) y =
x3 − x
x2 − 6x+ 5
(d) f(x) = − 3
x+ 2
(e) y =
4
x2 − 3x+ 2
(f) y =
1√
x+ 4
(g) y =
2x2√
x2 − 16
(h) f(x) = e1/x
(i) y = ln(x)
(j) y = tg(x)
6. Usando os limites fundamentais, calcule os limites dados:
(a) lim
x→0
sen(9x)
x
(b) lim
x→0
sen(4x)
3x
(c) lim
x→0
sen(6x)
sen(4x)
(d) lim
x→0
tg(ax)
x
(e) lim
x→0
1− cos(x)
x
(f) lim
x→0
cos(2x)− cos(3x)
x2
(g) lim
n→+∞
(
2n+ 3
2n+ 1
)n+1
(h) lim
x→ 3pi
2
(1 + cos(x))1/ cos(x)
(i) lim
x→2
10x−2 − 1
x− 2
(j) lim
x→2
5x − 25
x− 2
(k) lim
x→pi
2
(
1 +
1
tg(x)
)tg(x)
(l) lim
x→+∞
(
1 +
10
x
)x
(m) lim
x→−3
4x+3/5 − 1
x+ 3
(n) lim
x→1
3x−1/4 − 1
sen[5(x− 1)]
Respostas:
1. (a) (i) -3; (ii) -3
(b) (i) 1/2; (ii) 1/2
(c) (i) -5/3 (ii) -5/3
(d) (i) 2/5; (ii) 2/5
(e) (i) 0; (ii) 0
(f) (i) 7; (ii) 7
(g) (i) 0; (ii) 0
(h) (i) -2/3; (ii) -2/3
(i) (i) 1; (ii) 1/3
2. (a) +∞
(b) 1/2
(c) +∞
(d) +∞
(e) -1/2
(f)
3
√
3/2
(g) 1/2
(h) +∞
(i) +∞
(j) 0
(k) 0
(l) +∞
(m) 3
(n) −∞
(o) -1/2
3. Não há resposta única.
4. Não há resposta única.
5. a) y = 1; x = −4
b) y = 2; x = −2; x = 1
c) x = 5
d) y = 0; x = −2
e) y = 0; x = 2; x = 1
f) y = 0; x = −4
g) x = 4; x = −4
h) y = 1; x = 0
i) x = 0
j) y = −1; x = 2k + pi2 , para n =
0,±1,±2,±3, . . .
6. a) 9
b) 4/3
c) 3/2
d) a
e) 0
f) 5/2
g) e
h) e
i) ln(10)
j) 23 ln(5)
k) e
l) e10
m)
2
5 ln(2)
n)
ln(3)
20
2