Cálculo I   2018.1   Lista 8
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Cálculo I 2018.1 Lista 8


DisciplinaCálculo I74.304 materiais1.348.962 seguidores
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Universidade do Estado do Rio de Janeiro
Instituto de Matema´tica e Estat´\u131stica
Departamento de Ana´lise Matema´tica
Disciplina: CA´LCULO DIFERENCIAL E INTEGRAL I
OITAVA LISTA DE EXERCI´CIOS
1) Usando a Regra de L\u2019 Ho\u2c6pital, prove que:
(a) lim
x\u21920
ex \u2212 e\u2212x
senx
= 2
(b) lim
x\u21920
x\u2212 arcsenx
sen3 x
= \u22121
6
(c) lim
x\u2192+\u221e
x2
ex
= 0
(d) lim
x\u2192+\u221e x
3e\u2212x = 0
(e) lim
x\u21920+
(senx)(lnx) = 0
(f) lim
x\u2192\u22121
4x3 + x2 + 3
x5 + 1
= 2
(g) lim
x\u2192pi
cos3 x2
senx
= 0
(h) lim
x\u21920+
xsenx = 1
(i) lim
x\u2192+\u221e
\uf8eb\uf8ed1 + 3
x
\uf8f6\uf8f8x = e3
(j) lim
x\u21920+
xe1/x = +\u221e
(k) lim
x\u21920
3x \u2212 10x
senx
= ln
3
10
(l) lim
x\u21920+
arctg x · sec
(pi
2
\u2212 x
)
= 1
(m) lim
x\u21920+
\uf8ee\uf8f01
x
+ lnx
\uf8f9\uf8fb =\u221e
(n) lim
x\u2192\u221e [x\u2212
3
\u221a
x3 \u2212 x] = 0
(o) lim
x\u2192+\u221e
x2 + 4
8x
= 0
(p) lim
x\u2192+\u221e
e3x
lnx
= +\u221e
(q) lim
x\u21920
4x\u2212 sen4x
x3
=
32
3
(r) lim
x\u2192+\u221e
x2 \u2212 9x + 4
3x2 + 7x + 8
=
1
3
(s) lim
x\u219213
ln(x\u2212 12)
x\u2212 13 = 1
(t) lim
x\u2192+\u221e x(e
2/x \u2212 1) = 2
(u) lim
x\u2192+\u221e xtg
7
x
= 7
(v) lim
x\u21920+
(1\u2212 cos x) lnx = 0
(w) lim
x\u21920+
x
2
2+lnx = e2
(x) lim
x\u21920+
(ex + x)1/x = e2
(y) lim
x\u2192+\u221e
ln
(
1 + 1x
)
sen
(
1
x
) = 1
(z) lim
x\u2192pi/2
tgx · ln
\uf8eb\uf8ed 1
senx
\uf8f6\uf8f8 = 0
1
2) Determine os valores de a e b para que a func¸a\u2dco
f (x) =
\uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3
(coshx + ax)b/x , x > 0
e , x = 0
ax + b , x < 0
seja cont´\u131nua para todos os nu´meros reais. R: a = 1/e, b = e
3) Calcule:
(a) lim
x\u21920+
(x + 1)cotg x
(b) lim
x\u21920
cosx
6x\u2212 2
(c) lim
x\u21921
1\u2212 x + ln x
x3 \u2212 3x + 2
(d) lim
x\u2192+\u221e
ln(2 + ex)
3x
(e) lim
x\u21920
\uf8eb\uf8ed 1
x2 + x
\u2212 1
cosx\u2212 1
\uf8f6\uf8f8
(f) lim
x\u2192\u22122
10 + 3x\u2212 x2
2x2 + 12x + 16
(g) lim
x\u21920+
x2
x\u2212 senx
R:
(a) e
(b) \u22121
2
(c) \u22121
6
(d)
1
3
(e) +\u221e
(f)
7
4
(g) +\u221e
2