Lista 3 - Limites

Prévia do material em texto

Universidade Federal de Sergipe -UFS
Departamento de Arquitetura e Urbanismo - DAU
Professor: Rafael Oliveira
Lista 3
Ca´lculo I.
1. Calcule os limites abaixo:
(a) lim
x→2
x2 + x− 6
x− 2
(b) lim
x→4
x2 − 4x
x2 − 3x− 4
(c) lim
x→1
x3 − 1
x2 − 1
(d) lim
x→−2
3x4 + 2x2 − x+ 1
(e) lim
x→√2
x4 − 3x2 − 1
(f) lim
x→ 3√2
x5 − x3 + x2 − 1
(g) lim
x→9
9− x
3−√x
(h) lim
t→0
(2 + t)3 − 8
t
(i) lim
x→0
(
1
x
− 1
x2 + x
)
(j) lim
x→0
(3 + x)−1 − 3−1
x
(k) lim
x→1
√
x− x2
1−√x
(l) lim
x→9
x2 − 81√
x− 3
(m) lim
x→0
|x|
(n) lim
x→0
|x|
x
(o) lim
x→−4
|x+ 4|
(p) lim
x→2
|x− 2|
x− 2
(q) lim
x→√−4−+
|x+ 4|
x+ 4
(r) lim
x→√pi+
|x−√pi|
x−√pi
(s) lim
x→0
(
1
x
− 1|x|)
(t) lim
x→2+
x+ 2
x2 − 4
(u) lim
x→0−
√
3 + x2
x
(v) lim
x→2+
x+ 2
x2 − 4
(w) lim
x→3+
√
x2 − 9
x− 3
(x) lim
x→−2−
x+ 1
x2 − 4
(y) lim
x→−7−
x+ 2
x+ 7
(z) lim
x→4−
√
16− x2
x− 4
2. Ache os limites abaixo:
(a) lim
x→+∞
2x+ 1
5x− 2
(b) lim
x→−∞
2x+ 7
4− 5x
(c) lim
x→+∞
7x2 − 2x+ 1
3x2 + 8x+ 5
(d) lim
x→+∞
2x2 − 3x
x+ 1
(e) lim
x→−∞
2x3 − 4
5x+ 3
(f) lim
x→+∞
2x2 + 1
5x3 − 2x2 + 1
(g) lim
x→+∞
√
x2 + 2
x+ 4
(h) lim
x→−∞
√
x2 − 2x+ 3
x+ 5
(i) lim
x→−∞
√
x4 + 1
2x2 − 3
(j) lim
x→+∞
(
√
x2 + 1− x)
(k) lim
x→+∞
(
√
3x2 + x− 2x)
2
3. Seja
f(x) =

x se x < 0;
8− x se x > 2;
x2 se 0 < x ≤ 2.
(a) Calcule os limites abaixo:
i. lim
x→0−
f(x)
ii. lim
x→0+
f(x)
iii. lim
x→1−
f(x)
iv. lim
x→1+
f(x)
v. lim
x→2−
f(x)
vi. lim
x→2+
f(x)
(b) Esboc¸e o gra´fico de f .
4. Seja
g(x) =

x2 − x− 2
x+ 1
se x < −1;
x2 − x− 2
x− 2 se −1 ≤ x < 2;
x2
x3 − 4 se x = 2;
x2 − x− 2
(x+ 1)(x− 2) se x > 2.
(a) Calcule os limites abaixo:
i. lim
x→−1−
g(x)
ii. lim
x→−1+
g(x)
iii. lim
x→2−
g(x)
iv. lim
x→2+
g(x)
v. lim
x→−1
g(x)
vi. lim
x→2
g(x)
(b) Esboc¸e o gra´fico de g.
5. Use o teorema do confronto para calcular os seguintes limites:
(a) lim
x→0
x.sen(
1
x
)
(b) lim
x→0+
√
x.esen(
pi
x
)
3
6. Ache os valor dos seguintes limites:
(a) lim
x→0
1− cos x
sen x
(b) lim
x→0
2 tag2 x
x2
(c) lim
x→0
x2 − 1
sen(x2 − 1)
(d) lim
x→0
tag x
sen 2x
7. Trac¸e um esboc¸o do gra´fico das seguintes func¸o˜es:
(a) f(x) =
5x2
x2 − 4
(b) g(x) =
2x+ 1
x− 3
(c) h(x) =
2x
6x2 + 11x− 10
(d) l(x) =
4x2
x2 − 9
(e) j(x) =
x2
4− x2
4