Exercícios complementares_Unidade 2

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1 
 
 
UNIVERSIDADE FEDERAL DE SANTA CATARINA 
CAMPUS DE JOINVILLE 
CURSO DE ENGENHARIA DA MOBILIDADE 
EMB 5001 – CÁLCULO DIFERENCIAL E INTEGRAL I 
 
Professores: Alexandre Mikowski e Rafael Machado Casali 
 
EXERCÍCIOS COMPLEMENTARES 
 
Unidade 2 – Noções sobre limite e continuidade 
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1 – Seja )(xf a função definida pelo gráfico: 
 
Intuitivamente, encontre se existir: 
(a) )(lim
3
xf
x −→
 
(b) )(lim
3
xf
x +→
 
(c) )(lim
3
xf
x→
 
(d) )(lim xf
x −∞→
 
(e) )(lim xf
x +∞→
 
(f) )(lim
4
xf
x→
 
 
 
 
 
 
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 2 
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2 – Seja )(xf a função definida pelo gráfico: 
 Intuitivamente, encontre se existir: 
(a) )(lim
2
xf
x +−→
 
(b) )(lim
2
xf
x −−→
 
(c) )(lim
2
xf
x −→
 
(d) )(lim xf
x +∞→
 
 
 
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3 – Seja )(xf a função definida pelo gráfico: 
 
Intuitivamente, encontre se existir: 
(a) )(lim
0
xf
x +→
 
(b) )(lim
0
xf
x −−→
 
(c) )(lim
0
xf
x→
 
(d) )(lim xf
x +∞→
 
(e) )(lim xf
x −∞→
 
(f) )(lim
2
xf
x→
 
 
 
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 3 
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4 – Seja )(xf a função definida pelo gráfico: 
 Intuitivamente, encontre se existir: 
(a) )(lim
2
xf
x +→
 
(b) )(lim
2
xf
x −→
 
(c) )(lim xf
x +∞→
 
(d) )(lim xf
x −∞→
 
(e) )(lim
1
xf
x→
 
 
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5 – Seja )(xf a função definida pelo gráfico: 
 Intuitivamente, encontre se existir: 
(a) )(lim
1
xf
x +→
 
(b) )(lim
1
xf
x −→
 
(c) )(lim
1
xf
x→
 
(d) )(lim xf
x +∞→
 
(e) )(lim xf
x −∞→
 
 
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 4 
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6 – É dado Lxf
ax
=
→
)(lim . Determinar um número δ para o ε dado tal que Lxf −)( < ε sempre que 
0 < ax − < δ. 
(a) 01,0,8)42(lim
2
==+
→
εx
x
 
(b) 5,0,10)73(lim
1
==+−
−→
εx
x
 
(c) 1,0,4
2
4lim
2
2
=−=
+
−
−→
ε
x
x
x
 
(d) 75,0,2
1
1lim
2
5
==
−
−
→
ε
x
x
x
 
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7 – Calcular os limites abaixo usando as propriedades de Limites. 
(a) )573(lim 2
0
xx
x
−−
→
 
(b) )273(lim 2
3
+−
→
xx
x
 
(c) )26(lim 45
1
++−
−→
xx
x
 
(d) )72(lim
2
1
+
→
x
x
 
(e) [ ]13
1
)2.()4(lim −
−→
++ xx
x
 
(f) [ ])2.()2(lim 10
0
+−
→
xx
x
 
(g) 
13
4lim
2
−
+
→ x
x
x
 
(h) 
2
3lim
2 +
+
→ t
t
t
 
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 5 
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(i) 
1
1lim
2
1
−
−
→ x
x
t
 
(j) 
2
65lim
2
2 +
++
→ t
tt
t
 
(k) 
2
65lim
2
2
−
+−
→ t
tt
t
 
(l) 
s
s
s 2
4lim
2
1
+
→
 
(m) 3
4
32lim +
→
x
x
 
(n) ( ) 32
7
23lim +
→
x
x
 
(o) 
x
xx
x 3
2lim
2
2
−
→
 
(p) 
43
2lim
2
−
−
→ x
xx
x
 
(q) [ ]xxx
x
cotgcossen2lim
2
+−
→pi
 
(r) ( )xe x
x
4lim
4
+
→
 
(s) ( ) 41
3
1
32lim +
−→
x
x
 
(t) 
4
senhlim
2
x
x→
 
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8 – Seja 





>−
≤−
=
.3,73
3,1)(
xx
xx
xf
. Esboçar o gráfico de )(xf . 
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 6 
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Calcule: 
(a) )(lim
3
xf
x −→
 
(b) )(lim
3
xf
x +→
 
(c) )(lim
3
xf
x→
 
(d) )(lim
5
xf
x −→
 
(e) )(lim
5
xf
x +→
 
(f) )(lim
5
xf
x→
 
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9 – Seja 





=
≠+−
=
.3,7
3,12)(
2
x
xxx
xh
 
Calcule o )(xh
x 3
lim
→
. Esboce o gráfico de h(x). 
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10 – Seja 





=
≠
−
−
=
.3,0
3,
3
3
)(
x
x
x
x
xg
 
(a) Esboce o gráfico de )(xg . 
(b) Achar, se existirem ).(lime)(lim),(lim
333
xgxgxg
xxx →→→ −+
 
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11 – Verifique se 
1
1lim
1
−
→ xx
 existe. 
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 7 
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12 – Seja )5(
)25()(
2
−
−
=
x
x
xf . 
Calcule os limites indicados se existirem: 
(a) )(lim
0
xf
x→
 
(b) )(lim
5
xf
x +→
 
(c) )(lim
5
xf
x +−→
 
(d) )(lim
5
xf
x→
 
(e) )(lim
5
xf
x −→
 
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13 – Calcule os limites: 
(a) 
1
1lim 2
3
1
−
+
−→ x
x
x
 
(b) )3)(2(
44lim
23
2
−+
++
−→ tt
ttt
t
 
(c) 
253
103lim 2
2
2
−−
−+
→ xx
xx
x
 
(d) 52
532lim
2
2
5
−
−−
→ t
tt
t
 
(e) 
ax
axax
ax
−
−−+
→
)1(lim
2
 
(f) 
36254
20173lim 2
2
4 +−
+−
→ xx
xx
x
 
(g) 
43
56lim 2
2
1
−−
++
−→ xx
xx
x
 
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 8 
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(h) 
23
1lim 2
2
1 ++
−
−→ xx
x
x
 
(i) 
2
4lim
2
2
−
−
→ x
x
x
 
(j) 
2012
65lim 2
2
1 +−
+−
−→ xx
xx
x
 
(k) 
h
h
h
16)2(lim
4
0
−+
→
 
(l) 
t
t
t
16)4(lim
2
0
−+
→
 
(m) 
t
t
t
5325lim
0
−+
→
 
(n) 0,lim
2
0
>
−+
→
a
t
abta
t
 
(o) 
1
1lim
1
−
−
→ h
h
h
 
(p) 
4
)8(2
lim
2
4 +
+−
−→ h
hh
h
 
(q) 
h
h
h
28lim
3
0
−+
→
 
(r) 
x
x
x
−
−+
→
11lim
0
 
(s) 0,,lim
22
22
0
>
−+
−+
→
ba
bbx
aax
x
 
(t) 0,lim
33
≠
−
−
→
a
ax
ax
ax
 
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 9 
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(u) 
1
1lim
4
3
1
−
−
→ x
x
x
 
(v) 2
33 2
1 )1(
12lim
−
+−
→ x
xx
x
 
(w) 
x
x
x +−
+−
→ 51
53lim
4
 
(x) 
x
xx
x
−−+
→
11lim
0
 
____________________________________________________________________________________________14 – Se 
xx
xx
xf
57
3)(
−
+
= , calcule: 
(a) )(lim xf
x +∞→
 
(b) )(lim xf
x −∞→
 
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15 – Se 2)2(
1)(
+
=
x
xf , calcule: 
(a) )(lim
2
xf
x −→
 
(b) )(lim xf
x +∞→
 
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16 – Calcule os limites: 
(a) )143(lim 23 −+
+∞→
xx
x
 
(b) 





+−
+∞→ 2
412lim
xxx
 
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 10 
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(c) 
1
1lim 2 +
+
+∞→ t
t
t
 
(d) 
1
1lim 2 +
+
−∞→ t
t
t
 
(e) 
352
32lim 2
2
−+
+−
+∞→ tt
tt
t
 
(f) 
7
232lim 2
35
+−
+−
+∞→ x
xx
x
 
(g) 2
25
2
73lim
x
xx
x
−
+−
−∞→
 
(h) 
37
25lim 3
3
+
+−
−∞→ x
x
x
 
(i) 
x
xx
x
13lim
2 ++
+∞→
 
(j) 3
103lim
x
xxx
x
−+
+∞→
 
(k) 
1
1lim
2
+
+
+∞→ x
x
x
 
(l) 
13
4310lim 2
2
−
+−
+∞→ x
xx
x
 
(m) 
1
12lim 2
3
−
+−
−∞→ x
xx
x
 
(n) 
1
12lim 2
3
−
+−
−∞→ x
xx
x
 
(o) 
1
15lim 34
23
+−+
−+−
−∞→ xxx
xxx
x
 
 
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 11 
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17 – Determinar as assíntotas horizontais e verticais do gráfico das seguintes funções: 
(a) 
4
4)(
−
=
x
xf
 
(b) 
2
3)(
+
−
=
x
xf
 
(c) 
23
4)( 2 +−= xxxf 
(d) )4)(3(
1)(
+−
−
=
xx
xf
 
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18 – Investigue a continuidade nos pontos indicados: 
(a) 






=
≠
=
0,0
0,sen
)(
x
x
x
x
xf
 em x = 0. 
(b) 






=
≠
−
=
2,3
2,
4
8-x
)(
2
3
x
x
xxf
 em x = 2. 
(c) ,
1
73)( 2
2
+
+−
=
x
xx
xf
 em x = 2. 
(d) ,
33
2)( 32
−−+
=
xxx
xf
 em x = -3. 
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19 – Calcule os limites aplicando os limites fundamentais. 
(a) 
x
x
x
)9(sen
0
lim
→
 
(b) 
x
x
x 3
)4(senlim
0→
 
 12 
(c) )7(sen
)10(senlim
0 x
x
x→
 
(d) 
x
ax
ox
)(tglim
→
 
(e) 3
3
0
)2/(senlim
x
x
x→
 
(f) 20
cos1lim
x
x
x
−
→
 
(g) 
x
x
x
cos1lim
0
−
→
 
(h) )4(sen32
)2(sen6lim
0 xx
xx
x +
−
→
 
(i) 
x
x x






+
+∞→
21lim
 
(j) 
x
x x
x






++∞→ 1
lim
 
(k) 
1
12
32lim
+
+∞→






+
+
n
x n
n
 
(l) 





−
→ x
e
x
x
1lim
2
0
 
(m) 





−
−
→ 2
255lim
0 x
x
x
 
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Observações: 
I. Os exercícios propostos foram selecionados do livro: FLEMING, D. M. & GONÇALVES, M. B. Cálculo A. 
Vol. 1; 6ª edição, Pearson Prentice Hall, São Paulo, 2007. Capítulo 3. 
II. A lista de exercícios corresponde aos tópicos da ementa: Noções sobre limite e continuidade. 
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