Cálculo I 2018.1 Lista 8

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Universidade do Estado do Rio de Janeiro
Instituto de Matema´tica e Estat´ıstica
Departamento de Ana´lise Matema´tica
Disciplina: CA´LCULO DIFERENCIAL E INTEGRAL I
OITAVA LISTA DE EXERCI´CIOS
1) Usando a Regra de L’ Hoˆpital, prove que:
(a) lim
x→0
ex − e−x
senx
= 2
(b) lim
x→0
x− arcsenx
sen3 x
= −1
6
(c) lim
x→+∞
x2
ex
= 0
(d) lim
x→+∞ x
3e−x = 0
(e) lim
x→0+
(senx)(lnx) = 0
(f) lim
x→−1
4x3 + x2 + 3
x5 + 1
= 2
(g) lim
x→pi
cos3 x2
senx
= 0
(h) lim
x→0+
xsenx = 1
(i) lim
x→+∞
1 + 3
x
x = e3
(j) lim
x→0+
xe1/x = +∞
(k) lim
x→0
3x − 10x
senx
= ln
3
10
(l) lim
x→0+
arctg x · sec
(pi
2
− x
)
= 1
(m) lim
x→0+
1
x
+ lnx
 =∞
(n) lim
x→∞ [x−
3
√
x3 − x] = 0
(o) lim
x→+∞
x2 + 4
8x
= 0
(p) lim
x→+∞
e3x
lnx
= +∞
(q) lim
x→0
4x− sen4x
x3
=
32
3
(r) lim
x→+∞
x2 − 9x + 4
3x2 + 7x + 8
=
1
3
(s) lim
x→13
ln(x− 12)
x− 13 = 1
(t) lim
x→+∞ x(e
2/x − 1) = 2
(u) lim
x→+∞ xtg
7
x
= 7
(v) lim
x→0+
(1− cos x) lnx = 0
(w) lim
x→0+
x
2
2+lnx = e2
(x) lim
x→0+
(ex + x)1/x = e2
(y) lim
x→+∞
ln
(
1 + 1x
)
sen
(
1
x
) = 1
(z) lim
x→pi/2
tgx · ln
 1
senx
 = 0
1
2) Determine os valores de a e b para que a func¸a˜o
f (x) =

(coshx + ax)b/x , x > 0
e , x = 0
ax + b , x < 0
seja cont´ınua para todos os nu´meros reais. R: a = 1/e, b = e
3) Calcule:
(a) lim
x→0+
(x + 1)cotg x
(b) lim
x→0
cosx
6x− 2
(c) lim
x→1
1− x + ln x
x3 − 3x + 2
(d) lim
x→+∞
ln(2 + ex)
3x
(e) lim
x→0
 1
x2 + x
− 1
cosx− 1

(f) lim
x→−2
10 + 3x− x2
2x2 + 12x + 16
(g) lim
x→0+
x2
x− senx
R:
(a) e
(b) −1
2
(c) −1
6
(d)
1
3
(e) +∞
(f)
7
4
(g) +∞
2