A Quest˜ao 1. (3,0) a) Calcule, caso existam, os seguintes limites: i

A
Questão 1. (3,0) a) Calcule, caso existam, os seguintes limites:
√
i) lim+ √
3
x→2
x−2
√ = lim
x2 + 1 − 3 5 x→2+
√
x−2
= lim+
p
√ √
√
√
3
2 + 1)2 + 3 x2 + 1 3 5 + 3 52
(x
x−2
√ p
√
√
√
√
3
3
3
x2 + 1 − 3 5
(x2 + 1)2 + 3 x2 + 1 3 5 + 52
p
√
√
√
3
3
(x2 + 1)2 + 3 x2 + 1 3 5 + 52
√
= lim+
x→2
x→2
p
√
√
√
3
3
(x2 + 1)2 + 3 x2 + 1 3 5 + 52
(x − 2)(x + 2)
x→2
√
√
p
p
√
√
√
√
3
3
3
x − 2 ( 3 (x2 + 1)2 + 3 x2 + 1 3 5 + 52 )
(x2 + 1)2 + 3 x2 + 1 3 5 + 52
√
√
√
= lim+
= +∞,
x→2
x − 2 x − 2 (x + 2)
x − 2 (x + 2)
p
3
pois lim+
x−2
= lim+
x2 − 4
x→2
√
√
√
√
√
3
(x2 + 1)2 + 3 x2 + 1 3 5 + 52
3 3 25
=
x+2
4
e
1
3
√
x 2 + cos x + √
2x + cos x + 3 x
x
x
= lim
ii) lim
x→+∞
x→+∞
1
x + sen x
x 1 + sen x
x
1
3
pois lim
= lim √ = 0, |cos x| ≤ 1 e |sen x| ≤ 1.
x→+∞ x
x→+∞
x
1
lim+ √
= +∞
x→2
x−2
1
3
2 + cos x + √
2
x
x
= lim
= = 2,
1
x→+∞
1
1 + sen x
x
b) Verifique se a função
f (x) =




√
x2 − 2x + 1
, se x 6= 1
x2 − 3x + 2 + 2(x − 1) cos (x − 1)



1,
se x = 1
é contı́nua em x = 1. Justifique.
p
√
x2 − 2x + 1
(x − 1)2
lim f (x) = lim 2
= lim
x→1
x→1 x − 3x + 2 + 2(x − 1) cos (x − 1)
x→1 (x − 1)(x − 2) + 2(x − 1) cos (x − 1)
|x − 1|
x→1 (x − 1) [(x − 2) + 2 cos (x − 1)]
= lim
(x − 1)
1
lim+
= lim+
=1
x→1 (x − 1) [(x − 2) + 2 cos (x − 1)]
x→1 (x − 2) + 2 cos (x − 1)
−(x − 1)
−1
lim−
= lim−
x→1 (x − 1) [(x − 2) + 2 cos (x − 1)]
x→1 (x − 2) + 2 cos (x − 1)
=⇒ f não é contı́nua em x = 1.









= −1 

=⇒6 ∃ lim f (x)
x→1