Prova Cálculo 1 B Gabarito - Prof. Sacha Friedli

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❈á❧❝✉❧♦ ■✿ ✶❛ ♣r♦✈❛✱ ✶✶✴✵✹✴✷✵✶✺✱ ✽❤✵✵✱ ❉✉r❛çã♦✿ ✶❤✹✵✳ ❇
✶✳ (6♣ts) ❘❡s♦❧✈❛ ❛ s❡❣✉✐♥t❡ ❞❡s✐❣✉❛❧❞❛❞❡✿ x+ 3 > 4
x
✳
❙✉❜tr❛✐♥❞♦
4
x
❡♠ ❛♠❜♦s ❧❛❞♦s✱ ❝♦❧♦❝❛♥❞♦ ♥♦ ♠❡s♠♦ ❞❡♥♦♠✐♥❛❞♦r ❡ ❢❛t♦r❛♥❞♦✱ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ s❡ t♦r♥❛
(x+4)(x−1)
x
> 0
(3♣ts)✳ ▼♦♥t❛♥❞♦ ❛ t❛❜❡❧❛ ❞❡ s✐♥❛✐s ❞♦s três t❡r♠♦s x+ 4✱ x− 1 ❡ x ♦❜t❡♠♦s✿ S = (−4, 0) ∪ (1,+∞) (3♣ts)✳ ✭▲❡♠❜r♦ q✉❡
é ♠❡❧❤♦r ❡✈✐t❛r ♠✉❧t✐♣❧✐❝❛r ❡♠ ❛♠❜♦s ❧❛❞♦s ♣♦r x ❧á ♥♦ ✐♥í❝✐♦✱ ♣♦✐s ♣r❡❝✐s❛ s❛❜❡r s❡ ♠✉❞❛ ♦✉ ♥ã♦ ♦ s❡♥t✐❞♦ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡✱
❞❡♣❡♥❞❡♥❞♦ ❞❡ x s❡r ♣♦s✐t✐✈♦ ♦✉ ♥❡❣❛t✐✈♦✳✮
✷✳ (6♣ts) ▼♦♥t❡ ❡ ❝❛❧❝✉❧❡ ✉♠ ❧✐♠✐t❡ q✉❡ r❡♣r❡s❡♥t❡ ❛ ✐♥❝❧✐♥❛çã♦ ❞❛ r❡t❛ t❛♥❣❡♥t❡ ❛♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ f(x) = x3 ♥♦ ♣♦♥t♦
P = (1, 1)✳
❖ ❧✐♠✐t❡ ♣r♦❝✉r❛❞♦ é
lim
λ→1
f(λ)− f(1)
λ− 1 = limλ→1
λ3 − 1
λ− 1 , (3♣ts)
❡ é ❞❛ ❢♦r♠❛ ✏0/0✑✳ ❋❛③❡♥❞♦ ❛ ❞✐✈✐sã♦ ❞❡ λ3 − 1 ♣♦r λ− 1 ♦❜t❡♠♦s
lim
λ→1
λ3 − 1
λ− 1 = limλ→1(λ
2 + λ+ 1) = 3 .(3♣ts)
✸✳ (8♣ts) ❯s❛♥❞♦ s♦♠❡♥t❡ ♠ét♦❞♦s q✉❡ ❢♦r❛♠ ✈✐st♦s ❛té ❛❣♦r❛✱ ❝❛❧❝✉❧❡ limx→∞{
√
x2 + x− x}✱ limx→∞ x2−4x−5x2+5x+4 ✳
❖ ♣r✐♠❡✐r♦ ❧✐♠✐t❡ é ✐♥❞❡t❡r✐♠❛❞♦✱ ❞❛ ❢♦r♠❛ ✏∞−∞✑✳ ▼❛s✱ ♠✉❧t✐♣❧✐❝❛♥❞♦ ❡ ❞✐✈✐❞✐♥❞♦ ♣❡❧♦ ❝♦♥❥✉❣❛❞♦ ♦❜t❡♠♦s
lim
x→∞
{
√
x2 + x− x} = lim
x→∞
x√
x2 + x+ x
= lim
x→∞
1√
1 + 1
x
+ 1
(2♣ts) = 12 (2♣ts) .
❖ s❡❣✉♥❞♦ ❧✐♠✐t❡ é ❞❛ ❢♦r♠❛ ✏
∞
∞ ✑✳ ❈♦♠♦ ♦s t❡r♠♦s ♠❛✐s ✐♠♣♦rt❛♥t❡s sã♦ ♦s ❞❡ ❣r❛✉ 2✱ ❝♦❧♦❝❛♠♦s ❡❧❡s ❡♠ ❡✈✐❞ê♥❝✐❛✱
s✐♠♣❧✐✜❝❛♠♦s ♣♦r x2✱ ❡ ♦❜t❡♠♦s
lim
x→∞
x2 − 4x− 5
x2 + 5x+ 4
= lim
x→∞
1− 4
x
− 5
x2
1 + 5
x
+ 4
x2
= 11 = 1 .(4♣ts)
✹✳ (5♣ts) ❆ ❢✉♥çã♦ f é ❝♦♥tí♥✉❛ ❡♠ x = 0❄ ❏✉st✐✜q✉❡✳
f(x) =


e−
1
x
s❡ x > 0 ,
0 s❡ x = 0 ,
x
2
x2+1 s❡ x < 0 .
Pr❡❝✐s❛♠♦s ✈❡r s❡ limx→0 f(x) ❡①✐st❡ ❡ é ✐❣✉❛❧ ❛ f(0)✳ P♦r ✉♠ ❧❛❞♦✱
lim
x→0−
f(x) = lim
x→0−
x2
x+ 1
= 01 = 0 .(2♣ts)
P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠ ❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧ z = 1
x
✱ x→ 0+ ✐♠♣❧✐❝❛ z → +∞✱ ❡
lim
x→0+
f(x) = lim
x→0+
e−
1
x = lim
z→∞
e−z = 0 .(2♣ts)
▲♦❣♦ limx→0 f(x) = 0✱ ❡ ❝♦♠♦ f(0) = 0✱ f é ❝♦♥tí♥✉❛ ❡♠ x = 0✳(1♣ts)
✺✳ ✭❇❖◆❯❙✮ ✭♥♦ ♠á①✐♠♦ (5♣ts)✮ ❉ê ❛ ❞❡✜♥✐çã♦ r✐❣♦r♦s❛ ❞♦ s✐❣♥✐✜❝❛❞♦ ❞❡ ✏f(x) t❡♥❞❡r ❛ L q✉❛♥❞♦ x t❡♥❞❡ ❛ +∞✑✳ ❊♠
s❡❣✉✐❞❛✱ ✉s❛♥❞♦ ❛ ❞❡✜♥✐çã♦✱ ♠♦str❡ q✉❡ f(x) = 2x+3
x+1 t❡♥❞❡ ❛ 2 q✉❛♥❞♦ x t❡♥❞❡ ❛ +∞✳
❉✐③ s❡ q✉❡ ✏f(x) t❡♥❞❡ ❛ L q✉❛♥❞♦ x t❡♥❞❡ ❛ +∞✑ s❡ ♣❛r❛ t♦❞♦ ǫ > 0 ❡①✐st❡ ✉♠ ♥ú♠❡r♦ N t❛❧ q✉❡ f(x) − L ≤ ǫ ♣❛r❛
t♦❞♦ x ≥ N ✳ ❊s❝r❡✈❡✲s❡✿ limx→∞ f(x) = L✳ ◆♦ ❝❛s♦✱ ✜①❡♠♦s ✉♠ ǫ > 0 q✉❛❧q✉❡r ❡ ♣r♦❝✉r❡♠♦s s❛❜❡r s❡ | 2x+3x+1 − 2| ≤ ǫ ♣❛r❛
t♦❞♦ x s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✳ ❈♦♠♦ ✭❡❧♠❜r❡ q✉❡ s❡ tr❛t❛ ❞❡ ❝♦♥s✐❞❡r❛r x ♣♦s✐t✐✈♦✮ | 2x+3
x+1 − 2| = 1|x+1| = 1x+1 ≤ 1x ✱ ♣♦❞❡♠♦s
s✐♠♣❧❡s♠❡♥t❡ ♣❡❣❛r N = 1/ǫ✳
✷