Limites- Conteudo e Exercícios
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Limites- Conteudo e Exercícios


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x\u2212\u221aa\u221a
x2 \u2212 a2
(s) lim
x\u2192a
\u221a
x\u2212\u221aa+\u221ax\u2212 a\u221a
x2 \u2212 a2
(t) lim
x\u21921
x2 \u2212 x
2x2 + 5x\u2212 7
(u) lim
x\u2192\u22122
x3 + 8\u221a
x+ 2
5. Calcule os seguintes limites laterais:
(a) lim
x\u21920±
\u221a
1\u2212 cos(2x)
x
(b) lim
x\u21920±
cos(
pi
x
) (c) lim
x\u21920±
[[x]]
6. Verifique se os seguintes limites existem:
(a) lim
x\u21921
x3 \u2212 1
|x\u2212 1|
(b) lim
x\u21923
|x\u2212 3|
(c) lim
x\u21921
x2 \u2212 3x+ 2
x\u2212 1
(d) lim
x\u21925
x3 \u2212 6x2 + 6x\u2212 5
x2 \u2212 5x
(e) lim
x\u2192\u22124
x2 + 3x\u2212 4
x3 + 4x2 \u2212 3x\u2212 12
(f) lim
x\u21928
x\u2212 8
3
\u221a
x\u2212 2
160 CAPÍTULO 3. LIMITE E CONTINUIDADE DE FUNÇÕES
(g) lim
x\u21920
(cos(x)\u2212 [[sen(x)]])
(h) lim
x\u21920
(sen(x)\u2212 [[cos(x)]])
(i) lim
x\u21920+
x
a
\u2223\u2223 b
x
\u2223\u2223
(j) lim
x\u21920+
[[
x
a
]]
7. Calcule os seguintes limites no infinito:
(a) lim
x\u2192+\u221e
2x3 + 5x + 1
x4 + 5x3 + 3
(b) lim
x\u2192+\u221e
3x4 \u2212 2\u221a
x8 + 3x + 4
(c) lim
x\u2192\u2212\u221e
x2 \u2212 2x+ 3
3x2 + x+ 1
(d) lim
x\u2192+\u221e
x
x2 + 3x+ 1
(e) lim
x\u2192+\u221e
\u221a
x2 + 1
3x + 2
(f) lim
x\u2192\u2212\u221e
\u221a
x2 + 1
3x + 2
(g) lim
x\u2192+\u221e
\u221a
x+ 3
\u221a
x
x2 + 3
(h) lim
x\u2192+\u221e
(x\u2212
\u221a
x2 + 1)
(i) lim
x\u2192\u2212\u221e
3
\u221a
x
x2 + 3
(j) lim
x\u2192+\u221e
3
\u221a
x3 + 2x\u2212 1\u221a
x2 + x+ 1
(k) lim
x\u2192+\u221e
(
\u221a
x+ 1\u2212\u221ax+ 3)
(l) lim
x\u2192+\u221e
x5 + 1
x6 + 1
(m) lim
x\u2192+\u221e
x3 + x+ 1
3
\u221a
x9 + 1
(n) lim
x\u2192+\u221e
\u221a
x4 + 2
x3
(o) lim
x\u2192+\u221e
\u221a
x2
x3 + 5
(p) lim
x\u2192+\u221e
\u221a
x\u2212 1\u221a
x2 \u2212 1
(q) lim
x\u2192+\u221e
2x2 \u2212 x+ 3
x3 + 1
(r) lim
x\u2192+\u221e
3
\u221a
x2 + 8
x2 + x
(s) lim
x\u2192+\u221e
4x
x2 \u2212 4x + 3
(t) lim
x\u2192+\u221e
3x4 + x+ 1
x4 \u2212 5
(u) lim
x\u2192\u2212\u221e
x5 + x4 + 1
x6 + x3 + 1
(v) lim
x\u2192\u2212\u221e
x9 + 1
x9 + x6 + x4 + 1
(w) lim
x\u2192+\u221e
2x + 11\u221a
x2 + 1
(x) lim
x\u2192\u2212\u221e
6\u2212 7x
(2x + 3)4
8. Calcule os seguintes limites infinitos:
(a) lim
x\u2192+\u221e
x3 + 3x+ 1
2x2 + x+ 1
(b) lim
x\u21922+
x2 + 3x
x2 \u2212 4
(c) lim
x\u21921+
x3 \u2212 1
x2 \u2212 2x + 1
(d) lim
x\u2192+\u221e
(5\u2212 4x + x2 \u2212 x5)
(e) lim
x\u2192\u2212\u221e
5x3 \u2212 6x + 1
6x2 + x+ 1
(f) lim
x\u2192+\u221e
m
\u221a
x
(g) lim
x\u21923+
5
3\u2212 x
(h) lim
x\u21920+
2x + 1
x
(i) lim
x\u21921+
2x + 3
x2 \u2212 1
(j) lim
x\u21921\u2212
2x+ 3
x2 \u2212 1
(k) lim
x\u21923+
x2 \u2212 3x
x2 \u2212 6x + 9
(l) lim
x\u21922+
x2 \u2212 4
x2 \u2212 4x + 4
(m) lim
x\u21920+
sen(x)
x3 \u2212 x2
(n) lim
x\u21920+
ln(x)
x
(o) lim
x\u21920
ln(|x|)
(p) lim
x\u21920
tg(x)
x3
(q) lim
x\u2192pi
2
+
tg(x)
(r) lim
x\u21920
|x|
x3
sen(x)
3.9. EXERCÍCIOS 161
(s) lim
x\u2192 2
3
+
x2
4\u2212 9x2
(t) lim
x\u21920+
\u221a
x\u2212 1\u221a
x
(u) lim
x\u21921+
x\u2212 1\u221a
x\u2212 1
(v) lim
x\u2192 3
5
\u2212
1
5x\u2212 3
9. Se f(x) = 3x\u2212 5 e g(x) = x
2
\u2212 2
3
, calcule:
(a) lim
x\u21921
(f + g)(x)
(b) lim
x\u21921
(g \u2212 f)(x)
(c) lim
x\u21921
(g f)(x)
(d) lim
x\u21921
(f
g
)
(x)
(e) lim
x\u21921
( g
f
)
(x)
(f) lim
x\u21921
(f f)(x)
(g) lim
x\u21922
(f \u25e6 g)(x)
(h) lim
x\u21922
(g \u25e6 f)(x)
(i) lim
x\u2192\u2212 3
2
(f \u25e6 g \u25e6 f)(x)
(j) lim
x\u21922
ln(|f(x)|)
(k) lim
x\u2192 4
3
cos
( g(x)
f(x)
)
(l) lim
x\u21920
x sen
( 1
g(x)
)
(m) lim
x\u21920
x tg
( 1
g(x)
)
(n) lim
x\u21920
x cotg
( 1
g(x)
)
10. Calcule os seguintes limites:
(a) lim
x\u21920
sen(3x)
x
(b) lim
x\u21920
x2
sen(x)
(c) lim
x\u21920
tg(3x)
sen(4x)
(d) lim
x\u2192pi
2
1\u2212 sen(x)
2x\u2212 pi
(e) lim
x\u2192pi
sen(x)
x\u2212 pi
(f) lim
x\u2192+\u221e
x sen(
1
x
)
(g) lim
x\u21920
x\u2212 tg(x)
x+ tg(x)
(h) lim
x\u2192+\u221e
(1 +
2
x
)x+1
(i) lim
x\u21920
(
1 +
1
2x
)x
(j) lim
x\u21920
(1 + 2x)
1
x
(k) lim
x\u21920
e2x \u2212 1
x
(l) lim
x\u21920
ex
2 \u2212 1
x
(m) lim
x\u21920
5x \u2212 1
x
(n) lim
x\u21920
3x \u2212 1
x2
(o) lim
x\u21920
eax \u2212 ebx
sen(ax)\u2212 sen(bx) , a, b 6= 0
(p) lim
x\u21920
x cos2(x)
(q) lim
x\u21920
tg2(x)
x2 sec(x)
(r) lim
x\u2192+\u221e
(1\u2212 4
x
)x+4
(s) lim
x\u2192\u2212\u221e
(1\u2212 1
x
)x
11. Calcule lim
x\u2192a
f(x)\u2212 f(a)
x\u2212 a e limt\u21920
f(t+ a)\u2212 f(a)
t
, se:
(a) f(x) = x2, a = 2
(b) f(x) = x2 + 1, a = 2
(c) f(x) = 3x2 \u2212 x, a = 0
(d) f(x) = |x|2, a = 2
(e) f(x) =
\u221a
x, a = 1
(f) f(x) = x (1 \u2212 x), a = 1
(g) f(x) = cos(x), a = pi
(h) f(x) = (x\u2212 3)2, a = 1
(i) f(x) = ln(x), a = 1
(j) f(x) = e2x, a = 0
162 CAPÍTULO 3. LIMITE E CONTINUIDADE DE FUNÇÕES
12. Se |f(x)\u2212 f(y)| \u2264 |x\u2212 y|2, para todo x, y \u2208 R, verifique que: lim
x\u2192a
f(x)\u2212 f(a)
x\u2212 a = 0.
13. Verifique que lim
x\u2192+\u221e
(
\u221a
x+
\u221a
x\u2212
\u221a
x\u2212\u221ax) = 1.
14. No problema 51 do capítulo II, foi visto que o custo para remover x% de resíduos tóxicos
num aterro é dado por S(x) =
0.8x
100 \u2212 x , 0 < x < 100.
(a) Calcule lim
x\u2192100\u2212
S(x).
(b) Interprete o resultado obtido.
15. Suponha que 2000 reais são investidos a uma taxa de juros anual de 6% e os juros são
capitalizados continuamente.
(a) Qual é o saldo ao final de 10 anos? E de 50 anos?
(b) Que quantia deveria ser investida hoje a uma taxa anual de 7% de juros capitalizados
continuamente, de modo a se transformar, daqui a 20 anos, em 20000 reais?
16. Durante uma epidemia de dengue, o número de pessoas que adoeceram, num certo
bairro, após t dias é dado por L(t) =
100000
1 + 19900 e\u22120.8t
.
(a) Determine a quantidade máxima de indivíduos atingidos pela doença.
(b) Esboce o gráfico de L.
17. Esboce o gráfico das seguintes funções:
(a) y =
1
(x + 1) (x3 \u2212 1)
(b) y =
x
(x + 1) (x3 \u2212 1)
(c) y =
1
(x\u2212 1) (x3 + 1)
(d) y =
x
(x\u2212 1) (x3 + 1)
(e) y =
1
(x\u2212 3) (x + 2) (x2 + 1)
(f) y =
x2
(x\u2212 3) (x + 2) (x2 \u2212 1)
18. Use a continuidade da função para calcular os seguintes limites:
(a) lim
x\u2192pi
cos(x+ sen(x))
(b) lim
x\u21924
1 +
\u221a
x\u221a
x+ 1
(c) lim
x\u2192pi
2
e
1
sen(x)
(d) lim
x\u21921
1
arctg(x)
(e) lim
x\u21920
sen(x2 + sen(cos(x)))
x2 + 1
(f) lim
x\u21920
ln
( cos2(x) + 1\u221a
2 (x2 + 1)
)
3.9. EXERCÍCIOS 163
19. Verifique se as seguintes funções são contínuas:
(a) f(x) = argsenh(2x) (b) f(x) = cos(2x)
(c) f(x) =
x
x4 + 1
(d) f(x) = |sen(x)|
(e) f(x) = sec(x2 + 1) f) f(x) = tg(x2 + 1)
(g) f(x) =
{
2x se x \u2264 1
1 se x > 1
(h) f(x) =
\uf8f1\uf8f2
\uf8f3
x2 \u2212 4
x\u2212 2 se x 6= 2
4 se x = 2
Esboce os gráficos correspondentes.
20. Seja f(x) = x3 + x. Verifique que:
(a) |f(x)\u2212 f(2)| \u2264 20 |x\u2212 2| se 0 \u2264 x \u2264 3 (b) f é contínua em 2.
21. Determine o valor de L para que as seguintes funções sejam contínuas nos pontos dados:
(a) f(x) =
\uf8f1\uf8f2
\uf8f3
x2 \u2212 x
x
se x 6= 0
L se x = 0
, no ponto x = 0.
(b) f(x) =
\uf8f1\uf8f2
\uf8f3
x2 \u2212 9
x\u2212 3 se x 6= 3
L se x = 3
, no ponto x = 3.
(c) f(x) =
{
x+ 2L se x \u2265 \u22121
L2 se x < \u22121 , no ponto x = \u22121.
(d) f(x) =
{
4 3x se x < 0
2L + x se x \u2265 0 , no ponto x = 0.
(e) f(x) =
\uf8f1\uf8f2
\uf8f3
ex \u2212 1
x
se x 6= 0
L se x = 0
, no ponto x = 0.
(f) f(x) =
{
4\u2212 x+ x3 se \u2264 1
9\u2212 Lx2 se x > 1 , no ponto x = 1.
22. Verifique se as seguintes funções são contínuas.
(a) f(x) =
\uf8f1\uf8f2
\uf8f3
sen(x)
x
x 6= 0
0 x = 0 (b) f(x) =
\uf8f1\uf8f4\uf8f4\uf8f4\uf8f2
\uf8f4\uf8f4\uf8f4\uf8f3
|x2 \u2212 5x+ 6|
x2 \u2212 5x+ 6 x 6= 2, 3
1 x = 2
9 x = 3
164 CAPÍTULO 3. LIMITE E CONTINUIDADE DE FUNÇÕES
(c) f(x) =
\uf8f1\uf8f2
\uf8f3
1\u2212 x
1\u2212 x3 x 6= 1
1 x = 1
(d) f(x) =
\uf8f1\uf8f4\uf8f4\uf8f4\uf8f2
\uf8f4\uf8f4\uf8f4\uf8f3
1\u2212 x2 x < \u22121
ln(2\u2212 x2) \u22121 \u2264 x \u2264 1\u221a
x\u2212 1
x+ 1
x > 1
(e) f(x) =
\uf8f1\uf8f4\uf8f4\uf8f2
\uf8f4\uf8f4\uf8f3
1
5
(2x2 + 3) x \u2264 1
6\u2212 5x 1 < x < 3
x\u2212 3 x \u2265 3
(f) f(x) =
etg(x) \u2212 1
etg(x) + 1
(g) f(x) =
\uf8f1\uf8f2
\uf8f3
[[x + 3]] x < 0
(x + 1)3 \u2212 1
x
x > 0
23. Determine em que pontos as seguintes funções são contínuas:
(a) f(x) = arctg
(cos(x) + sen(x)
x4 + x2 + 1
)
(b) f(x) = cos(ln(
x4 + 4
x2
))
(c) f(x) =
x5 + x4 \u2212 x2 + 1
sec(x2 + 1)
(d) f(x) =
sen2(x2) + ln(x2 + 1)
x2 arctg(x)
(e) f(x) =
ex
2
+ esen(x) + 2
(x2 + 6)(ex + 1)
(f) f(x) =
cos([[x]])
[[x]]
24. Verifique se as seguintes equações admitem, pelo menos, uma raiz real:
(a) x3 + x2 \u2212 4x\u2212 15 = 0
(b) cos(x)\u2212 x = 0
(c) sen(x)\u2212 x+