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UNIVERSIDADE ESTADUAL DE MARING A´
ENGENHARIA MEC ANICAˆ
Lista de Exerc´ıcios No. 02
Limites e Continuidade
- Para a func¸ ˜ao f cujo gr´afico ´e dado abaixo, determine o solicitado em cada item, justifi-
cando quando n˜ao for poss´ıvel determinar.
x
y
1
1
(a) ( f ◦ f ◦ f )( 2 )
(b) lim x→-2+^
f (x)
(c) lim x→ 0 -^
f (x)
(d) lim x→ 0 +^
f (x)
(e) lim x→ 0
f (x)
(f) lim x→ 2 -^
f (x)
(g) lim x→ 2 +^
f (x)
(h) lim x→ 2
f (x)
(i) lim x→ 4 -^
f (x)
- Para a func¸ ˜ao f cujo gr´afico ´e dado abaixo, determine o solicitado em cada item, justifi-
cando quando n˜ao for poss´ıvel determinar.
x
y
1
1
(a) lim x→-7-^
f (x)
(b) lim x→-7+^
f (x)
(c) lim x→-
f (x)
(d) lim x→-3-^
f (x)
(e) lim x→-3+^
f (x)
(f) lim x→-
f (x)
(g) lim x→ 0 -^
f (x)
(h) lim x→ 0 +^
f (x)
(i) lim x→ 0
f (x)
(j) lim x→∞
f (x) (k) Ass´ıntotas horizontais (l) Ass´ıntotas verticais
- Para a func¸ ˜ao f cujo gr´afico ´e dado abaixo, determine o solicitado em cada item, justifi-
cando quando n˜ao for poss´ıvel determinar.
x
y
1
1
(a) ( f ◦ f )( 5 )
(b) lim x→-∞
f (x)
(c) lim x→-
f (x)
(d) lim x→-3-^
f (x)
(e) lim x→-3+^
f (x)
(f) lim x→-
f (x)
(g) lim x→-
f (x)
(h) lim x→-1-^
f (x)
(i) lim x→-1+^
f (x)
(j) lim x→ 0
f (x)
(k) lim x→ 3 -^
f (x)
(l) lim x→ 3 -^
f (x)
(m) lim x→ 3
f (x)
(n) lim x→ 4
f (x)
(o) lim x→ 5
f (x)
(p) lim x→∞
f (x)
(q) Ass´ıntotas horizontais
(r) Ass´ıntotas verticais
- Dado que
lim x→ 2
f (x) = 4 lim x→ 2
g(x) = − 2 lim x→ 2
h(x) = 0
encontre, se existir, o limite. Caso n˜ao exista, explique por quˆe.
(a) lim x→ 2
[ f (x) + 5 g(x)]
(b) lim x→ 2
[g(x) − 3 h(x)]
(c) lim x→ 2
f (x)g(x)h(x)
(d) lim x→ 2
h(x)
g(x)
(e) lim x→ 2
3 f (x)
g(x)
(f) lim x→ 2
g(x)h(x)
f (x)
(g) lim x→ 2
[(g(x))^3 + ( f (x))^2 ]
(h) lim x→ 2
3
f (x)g(x)
(i) lim x→ 2
g(x)
(j) lim x→ 2
e h(x)
(k) lim x→ 2
ln [g(x)]
(l) lim x→ 2
sen [g(x)π]
(m) lim x→ 2
tg
[
h(x)
f (x)
]
(n) lim x→ 2
arccos
[
g(x)
2
]
- Dada a func¸ ˜ao
f (x) =
2 x + 1 , se x ≤ 3
ax + b , se 3 ≤ x ≤ 5
x^2 + 2 se x ≥ 5
Determine os valores das constantes a e b para os quais os limites lim x→ 3
f (x) e lim x→ 5
f (x)
existam e construa o gr´afico de f nesse caso.
- Calcular lim x→ 1
( f + g)(x) se existir
f (x) =
5 x − 1 , se x ≤ 1
4 − x , se x > 1
f (x) =
3 − 2 x , se x < 1
x − 1 , se x = 1
4 x − 2 , se x > 1
- Calcule, o limite, quando existir. Caso n˜ao exista, explique por quˆe.
(a) lim x→ 2
x^2 + x − 6
x − 2
(b) lim x→ 2
x 2
|x − 2 |
(c) lim x→-
x^2 − 9
2 x^2 + 7 x + 3
(d) lim x→ 2
x^3 − x^2 − 8 x + 12
x^3 − x^2 − 16 x + 20
(e) lim x→-
x + 2
x^3 + 8
(f) lim x→ 0
9 + x − 3
x
(g) lim x→-
1 4 +^
1 x 4 + x
(h) lim x→ 0
1 + x −
1 − x
x
(i) lim x→ 16
x
16 x − x^2
(j) lim x→ 0
x
1 + x
x
(k) lim x→ 0
x √ 1 + 3 x − 1
(l) lim x→ 0. 5
2 x − 1
| 2 x^3 − x^2 |
(m) lim x→ 0
x
|x|
(n) lim x→ 0
x^3 + x^2 sen
π
x
(o) lim x→ 2
6 − x − 2 √ 3 − x − 1
(p) lim x→ 2
x^2 + 4 − 2
x^3 − 2 x^2 − 16 x + 32
(q) lim x→ 1
x^2 − 2 3
x + 1
(x − 1 )^2
(r) lim x→-
3 x + 5 + x + 3 3
x + 1 + 1
(s) lim x→ 0
[[ sen x ]]
(t) lim x→ 0
1 + x^3 −
1 + x^2
x^2
(u) lim x→ 0
x^4 + 1 −
1 + x^2
x^2
(v) lim x→ 0
3
x^2 + 1 − 4
1 − 2 x
x^2 + x
- Use o limite fundamental:
lim x→ 0
sen x
x
= 1 ou lim x→ 0
1 − cos x
x
= 0 ou lim x→ 0
1 − cos x
x^2
se necess´ario, para determinar os limites, caso existam
(a) lim x→ 0
sen(x^2 − 1 )
x − 1
(b) lim x→ 0
cos
π sen( 2 x)
3 x
(c) lim x→ 0
1 − cos( 3 x) − x sen^2 ( 3 x)
x
(d) lim x→ 0
tg(ax)
sen(bx)
, b, a ∈ R − { 0 }
(e) lim x→ 0
sen^2 (x + 2 ) − sen^2
x
(f) lim x→ 0
x 2 sen( 1 /x)
sen x
(g) lim x→ 0
x √ 1 − cos x
(h) lim x→ 0
1 + x sen x −
cos( 2 x)
tg^2 (x/ 2 )
(i) lim x→ 2 π/
1 − 256 cos 8 x
sen(x − π/ 3 )
(j) lim x→π/
6 x − sen( 2 x)
2 x + 3 sen( 4 x)
(k) lim x→ 0
1 − 2 cos x + cos( 2 x)
x^2
(l) lim x→n
n sen x − x sen n
n cos x − x cos n
(m) lim x→π/
sen 2 ( 6 x) + tg( 3 x)
3 x − π
- Encontre as ass´ıntotas verticais da func¸ ˜ao
f (x) =
x^2 + 1
3 x − 2 x^2
- Existe um n´umero a tal que
lim x→-
3 x 2
x^2 + x − 2
exista?. Caso exista, encontre a e o valor do limite.
- A figura mostra uma c´ırcunferˆencia fixa C 1 de equac¸ ˜ao (x − 1 )^2 + y^2 = 1 e um c´ırculo
C 2 , a ser escolhido, com raio r e centro na origem. P e o ponto´ ( 0 , r), Q e o ponto de´ intersec¸ ˜ao superior dos dois c´ırculos, e R e o ponto de intersec´ ¸ ˜ao da reta PQ com o eixo x. O que acontece com R quando C 2 se contrair?
x
y
C 2
P
Q
R
- A figura mostra uma c´ırcunferˆencia C de centro na origem e raio 1, e uma reta tangente
T a` C no ponto P. E ´e o ponto de intersec¸ ˜ao da reta T com o eixo y, D e o ponto´ ( 0 , 1 ) e PA ´e um segmento perpendicular ao eixo x
(a) lim x→∞
2 x + 3
(b) lim x→∞
1 − x − x 2
2 x^2 − 7
(c) lim x→∞
( 2 x^2 + 1 )^2
(x − 1 )^2 (x^2 + x)
(d) lim x→∞
9 x^6 − x
x^3 + 1
(e) lim x→∞
9 x^6 + x − 3 x)
(f) lim x→∞
x 4 − 3 x 2
x^3 − x + 2
(g) lim x→∞
x +
x +
x √ x + 1
(h) lim x→∞
8 x^3 + x^2 −
x^3 + x^2
x
(i) lim x→∞
x +
x +
x −
x
(j) lim x→∞
arctan(e x )
(k) lim x→∞
1 − ex
1 + 2 ex
(l) lim x→∞
(e−^2 x^ cos x
(m) lim x→∞
sen[π( 1 − x^2 )/( 1 + x^2 )]
x^4 sen( 1 /x)
