Cálculo I- Aula 3 Limites parte 2
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Cálculo I- Aula 3 Limites parte 2


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Cálculo I 
LIMITES \u2013 Parte 2
Professora: Mariah Rissi
LIMITES LATERAIS
\uf075 Quando examinamos
lim
\ud835\udc65\u2192\ud835\udc4e
\ud835\udc53(\ud835\udc65)
estamos pensando que \ud835\udc65 \u2192 \ud835\udc4e , isto é, \ud835\udc99 se aproxima de \ud835\udc82, por
valores maiores ou menores que \ud835\udc82.
\uf075 Entretanto, podemos fazer \ud835\udc99 se aproximar de \ud835\udc82 apenas por valores
maiores do que \ud835\udc82. Nesse caso, dizemos que \ud835\udc99 tende a \ud835\udc82 pela direita
e indicamos
lim
\ud835\udc65\u2192\ud835\udc4e+
\ud835\udc53(\ud835\udc65) .
\uf075 De modo análogo, podemos fazer \ud835\udc99 se aproximar de \ud835\udc82 apenas por
valores menores do que \ud835\udc82. Nesse caso, dizemos que \ud835\udc99 tende \ud835\udc82 pela
esquerda e indicamos
lim
\ud835\udc65\u2192\ud835\udc4e\u2212
\ud835\udc53(\ud835\udc65) .
LIMITES LATERAIS
LIMITES LATERAIS
EXEMPLO 1.
EXISTÊNCIA DE LIMITES 
OBSERVAÇÃO:
Todas as propriedades de limites permanecem válidas para limites laterais.
Exercícios
1. \ud835\udc53 \ud835\udc65 = \u124a\ud835\udc65
2 + 3 \ud835\udc60\ud835\udc52 \ud835\udc65 \u2265 1
5\ud835\udc65 \u2212 1 \ud835\udc60\ud835\udc52 \ud835\udc65 < 1
, calcule:
(a) lim
x\u21921+
f x = lim
x\u21921+
x2 + 3 =4
(b) lim
x\u21921\u2212
f(x) = lim
x\u21921\u2212
5x \u2212 1 =4
(c) lim
x\u21921
f(x) = lim
x\u21921
x2 + 3 = 4
Exercícios
2. \ud835\udc53 \ud835\udc65 = \u1250
\ud835\udc65 + 1 \ud835\udc60\ud835\udc52 \ud835\udc65 \u2265 2
2\ud835\udc65 \ud835\udc60\ud835\udc52 0 \u2264 \ud835\udc65 \u2264 2
\ud835\udc652 \ud835\udc60\ud835\udc52 \ud835\udc65 < 0
, calcule:
(a) lim
x\u21922+
f x = lim
x\u21922+
x + 1 =3
\ud835\udc4f lim
x\u21922\u2212
f(x) = lim
x\u21922\u2212
2x =4
(c) lim
x\u21922
f(x) = lim
x\u21922
x + 1 = \ud835\udc5bã\ud835\udc5c \ud835\udc52\ud835\udc65\ud835\udc56\ud835\udc60\ud835\udc61\ud835\udc52
(d) lim
x\u21920+
f(x) = lim
x\u21920+
2x = 0
(e) lim
x\u21920\u2212
f(x) = lim
x\u21920\u2212
x2 = 0
(f) lim
x\u21920
f(x) = lim
x\u21920
2x = 0
\ud835\udc65 + 12\ud835\udc65
\ud835\udc652
3. 
LIMITES NO INFINITOS
Definições Limites com \ud835\udc99 \u2192 ±\u221e \u2236
1. Dizemos que \ud835\udc53(\ud835\udc65) possui o limite \ud835\udc73 quando tende a mais infinito 
e escrevemos: 
lim
\ud835\udc65\u2192+\u221e
\ud835\udc53 \ud835\udc65 = \ud835\udc3f
se, à medida que \ud835\udc65 se distancia da origem no sentido positivo, 
\ud835\udc53(\ud835\udc65) fica cada vez mais próximo de \ud835\udc3f.
2. Dizemos que \ud835\udc53(\ud835\udc65) possui o limite \ud835\udc74 com \ud835\udc99 tendendo a menos 
infinito e escrevemos: 
lim
\ud835\udc65\u2192\u2212\u221e
\ud835\udc53 \ud835\udc65 = \ud835\udc40
se, à medida que \ud835\udc65 se distancia da origem no sentido negativo, 
\ud835\udc53(\ud835\udc65) fica cada vez mais próximo de \ud835\udc40. 
LIMITES NO INFINITOS
EXEMPLOS:
Função Exponencial
LIMITES NO INFINITOS
EXEMPLOS:
Função Logarítmica
LIMITES NO INFINITOS
\u221e \u221e
LIMITES NO INFINITOS
LIMITES NO INFINITOS
Limites de uma função Polinomial \ud835\udc91\ud835\udc82\ud835\udc93\ud835\udc82 \ud835\udc99 \u2192 ±\u221e
lim
±\u221e
\ud835\udc5d \ud835\udc65 = lim
±\u221e
\ud835\udc4e\ud835\udc5b\ud835\udc65
\ud835\udc5b .
Demonstração: 
lim
\ud835\udc65\u2192±\u221e
( \ud835\udc4e\ud835\udc5b\ud835\udc65
\ud835\udc5b + \u2026+ \ud835\udc4e2\ud835\udc65
2 + \ud835\udc4e1\ud835\udc65 + \ud835\udc4e0) =
= lim
\ud835\udc65\u2192±\u221e
\ud835\udc65\ud835\udc5b \ud835\udc4e\ud835\udc5b +
\ud835\udc4e\ud835\udc5b\u22121
\ud835\udc65
+
\ud835\udc4e\ud835\udc5b\u22122
\ud835\udc652
+ \u2026+
\ud835\udc4e1
\ud835\udc65\ud835\udc5b\u22121
+
\ud835\udc4e0
\ud835\udc65\ud835\udc5b
Mas: lim
\ud835\udc65\u2192±\u221e
\ud835\udc4e\ud835\udc5b\u22121
\ud835\udc65
= lim
\ud835\udc65\u2192±\u221e
\ud835\udc4e\ud835\udc5b\u22122
\ud835\udc652
= \u2026 = lim
\ud835\udc65\u2192±\u221e
\ud835\udc4e1
\ud835\udc65\ud835\udc5b\u22121
= lim
\ud835\udc65\u2192±\u221e
\ud835\udc4e0
\ud835\udc65\ud835\udc5b
= 0
Logo: 
lim
\ud835\udc65\u2192±\u221e
\ud835\udc65\ud835\udc5b \ud835\udc4e\ud835\udc5b + 0 + 0 + \u2026+ 0 + 0 = lim
±\u221e
\ud835\udc4e\ud835\udc5b\ud835\udc65
\ud835\udc5b.
LIMITES NO INFINITOS
Limites de uma função Polinomial \ud835\udc91\ud835\udc82\ud835\udc93\ud835\udc82 \ud835\udc99 \u2192 ±\u221e
lim
±\u221e
\ud835\udc5d \ud835\udc65
\ud835\udc5e(\ud835\udc65)
= lim
±\u221e
\ud835\udc4e\ud835\udc5b\ud835\udc65
\ud835\udc5b
\ud835\udc4f\ud835\udc5b\ud835\udc65
\ud835\udc5b
.
De forma análoga, 
Observações: * Para , temos: 
LIMITES NO INFINITOS
Exemplos:
1. Calcule, 
LIMITES NO INFINITOS
2. Calcule os limites a seguir:
a) 
lim
\ud835\udc65\u2192+\u221e
3\ud835\udc653 \u2212 2\ud835\udc652 + 2\ud835\udc65 \u2212 5
3 \u2212 \ud835\udc65 \u2212 2\ud835\udc652
= lim
\ud835\udc65\u2192+\u221e
3\ud835\udc653 \u2212 2\ud835\udc652 + 2\ud835\udc65 \u2212 5
\ud835\udc652
3 \u2212 \ud835\udc65 \u2212 2\ud835\udc652
\ud835\udc652
= lim
\ud835\udc65\u2192+\u221e
3\ud835\udc653
\ud835\udc652
\u2212
2\ud835\udc652
\ud835\udc652
+
2\ud835\udc65
\ud835\udc652
\u2212
5
\ud835\udc652
3
\ud835\udc652
\u2212
\ud835\udc65
\ud835\udc652
\u2212
2\ud835\udc652
\ud835\udc652
= lim
\ud835\udc65\u2192+\u221e
3\ud835\udc65 \u2212 2 +
2
\ud835\udc65 \u2212
5
\ud835\udc652
3
\ud835\udc652
\u2212
1
\ud835\udc65
\u2212 2
= lim
\ud835\udc65\u2192+\u221e
3\ud835\udc65 \u2212 2
\u22122
= lim
\ud835\udc65\u2192+\u221e
\u2212
(3\ud835\udc65 \u2212 2)
2
= lim
\ud835\udc65\u2192+\u221e
\u2212
3
2
\ud835\udc65 = \u2212
3
2
lim
\ud835\udc65\u2192+\u221e
\ud835\udc65
=\u2212
3
2
+\u221e =+\u221e
** O limite de cada fração do tipo 
\ud835\udc50
\ud835\udc65\ud835\udc5a
, com \ud835\udc65 \u2192 ±\u221e, é igual a zero!!
LIMITES NO INFINITOS
b)
c) 
lim
\ud835\udc65\u2192\u2212\u221e
3\ud835\udc652 \u2212 \ud835\udc65 + 2
5 \u2212 \ud835\udc652
= lim
\ud835\udc65\u2192\u2212\u221e
3\ud835\udc652 \u2212 \ud835\udc65 + 2
\ud835\udc652
5 \u2212 \ud835\udc652
\ud835\udc652
= lim
\ud835\udc65\u2192\u2212\u221e
3\ud835\udc652
\ud835\udc652
\u2212
\ud835\udc65
\ud835\udc652
+
2
\ud835\udc652
5
\ud835\udc652
\u2212
\ud835\udc652
\ud835\udc652
=
= lim
\ud835\udc65\u2192\u2212\u221e
3 \u2212
1
\ud835\udc65 +
2
\ud835\udc652
5
\ud835\udc652
\u2212 1
= lim
\ud835\udc65\u2192\u2212\u221e
3
\u22121
= \u22123
lim
\ud835\udc65\u2192+\u221e
2\ud835\udc65 + 3
3\ud835\udc652 + \ud835\udc65 \u2212 4
= lim
\ud835\udc65\u2192+\u221e
2\ud835\udc65 + 3
\ud835\udc652
3\ud835\udc652 + \ud835\udc65 \u2212 4
\ud835\udc652
= lim
\ud835\udc65\u2192+\u221e
2\ud835\udc65
\ud835\udc652
+
3
\ud835\udc652
3\ud835\udc652
\ud835\udc652
+
\ud835\udc65
\ud835\udc652
\u2212
4
\ud835\udc652
= lim
\ud835\udc65\u2192+\u221e
2
\ud835\udc65 +
3
\ud835\udc652
3 +
1
\ud835\udc65
\u2212
4
\ud835\udc652
= lim
\ud835\udc65\u2192+\u221e
0
3
= 0
Exercícios
LIMITES INFINITOS
lim
\ud835\udc65\u21922
\ud835\udc53 \ud835\udc65 = lim
\ud835\udc65\u21922
3
\ud835\udc65 \u2212 2 2
= +\u221e
LIMITES INFINITOS
lim
\ud835\udc65\u21922
\ud835\udc53 \ud835\udc65 = lim
\ud835\udc65\u21922
\u2212 3
\ud835\udc65 \u2212 2 2
= \u2212\u221e
LIMITES INFINITOS
\ud835\udc53 \ud835\udc65 =
1
\ud835\udc65
\ud835\udc53 \ud835\udc65 =
1
\ud835\udc652
LIMITES INFINITO (Propriedades)
LIMITES INFINITOS
Exemplo 1: Vamos calcular :lim
\ud835\udc65\u21921+
\ud835\udc65 + 1
1 \u2212 \ud835\udc65
1º) Achar raiz de \ud835\udc65 + 1 e fazer estudo do 
sinal:
\ud835\udc65 + 1 = 0
\ud835\udc65 = \u22121
\u22121
+ 
-
2º) Achar raiz de 1 \u2212 \ud835\udc65 e fazer estudo do 
sinal:
1 \u2212 \ud835\udc65 = 0
\u2212\ud835\udc65 = \u22121 × (\u22121)
\ud835\udc65 = 1
+ 
-
3º) Análise das funções:
-1 1
- + + 
+ + -
- + -
1
\ud835\udc3c
\ud835\udc3c\ud835\udc3c
\ud835\udc3c
\ud835\udc3c\ud835\udc3c
1+
Logo, = \u2212\u221elim\ud835\udc65\u21921+
\ud835\udc65 + 1
1 \u2212 \ud835\udc65
LIMITES INFINITOS
Exemplo 2: Vamos calcular :lim
\ud835\udc65\u21921
4
\ud835\udc65 \u2212 1 2
1º) Fazer estudo do sinal com a constante 4:
4
2º) Achar raiz de \ud835\udc65 \u2212 1 e fazer estudo do sinal:
3º) Análise das funções:
1 4
+ + +
- + +
- + +
\ud835\udc3c
\ud835\udc3c\ud835\udc3c
\ud835\udc3c\ud835\udc3c\ud835\udc3c
\ud835\udc3c
(\ud835\udc3c\ud835\udc3c)(\ud835\udc3c\ud835\udc3c\ud835\udc3c)
Logo, = +\u221e
+ +
\ud835\udc65 \u2212 1 = 0
\ud835\udc65 = 1
Obs.: Como o denominador é \ud835\udc65 \u2212 1 2 .
Consideramos como \ud835\udc65 \u2212 1 \ud835\udc65 \u2212 1 , logo
o processo no estudo dos sinais terá duas
linhas com o estudo do sinal da função
\ud835\udc65 \u2212 1;
\ud835\udc3c
\ud835\udc3c\ud835\udc3c
&
\ud835\udc3c\ud835\udc3c\ud835\udc3c + + +
lim
\ud835\udc65\u21921
4
\ud835\udc65 \u2212 1 2
Exercícios
1. Calcule os Limites a seguir:
CONTINUIDADE 
Informalmente dizemos que uma função é continua quando seu
gráfico não apresenta interrupções, ou seja, seu gráfico pode ser
traçado sem que o lápis se afaste do papel. Assim, para que uma
função \ud835\udc53 seja contínua em um ponto \ud835\udc65 = \ud835\udc4e é necessário que a função
esteja bem definida em \ud835\udc4e e que os valores de \ud835\udc53(\ud835\udc65), para \ud835\udc65 próximos
de \ud835\udc4e, esteja próximos de \ud835\udc53(\ud835\udc4e). Uma definição formal é dada a
seguir:
\ud835\udc53 é \ud835\udc51\ud835\udc52\ud835\udc53\ud835\udc56\ud835\udc5b\ud835\udc56\ud835\udc51\ud835\udc4e \ud835\udc5b\ud835\udc5c \ud835\udc5d\ud835\udc5c\ud835\udc5b\ud835\udc61\ud835\udc5c \ud835\udc4e;
\ud835\udc52\ud835\udc65\ud835\udc56\ud835\udc60\ud835\udc61\ud835\udc52 \ud835\udc5c \ud835\udc59\ud835\udc56\ud835\udc5a\ud835\udc56\ud835\udc61\ud835\udc52 \ud835\udc51\ud835\udc52 \ud835\udc53 \ud835\udc65 \ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc5b\ud835\udc51\ud835\udc5c \ud835\udc65 \u2192 \ud835\udc4e;
\ud835\udc5c \ud835\udc59\ud835\udc56\ud835\udc5a\ud835\udc56\ud835\udc61\ud835\udc52 \ud835\udc51\ud835\udc52 \ud835\udc53 \ud835\udc65 \ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc5b\ud835\udc51\ud835\udc5c \ud835\udc65 \u2192 \ud835\udc4e é \ud835\udc56\ud835\udc54\ud835\udc62\ud835\udc4e\ud835\udc59 \ud835\udc4e \ud835\udc53 \ud835\udc4e ;
CONTINUIDADE 
Exemplos
Verifique a continuidade das funções a seguir, nos pontos indicados:
1) \ud835\udc53 \ud835\udc65 =
1
\ud835\udc65
; \ud835\udc65 = 0.
\ud835\udc4e) \u2204\ud835\udc53 0 , \ud835\udc3f\ud835\udc5c\ud835\udc54\ud835\udc5c \ud835\udc4e \ud835\udc53\ud835\udc62\ud835\udc5bçã\ud835\udc5c \ud835\udc5bã\ud835\udc5c é \ud835\udc50\ud835\udc5c\ud835\udc5b\ud835\udc61í\ud835\udc5b\ud835\udc62\ud835\udc4e \ud835\udc5b\ud835\udc5c \ud835\udc5d\ud835\udc5c\ud835\udc5b\ud835\udc61\ud835\udc5c 0;
CONTINUIDADE 
Exemplos
2) \ud835\udc53 \ud835\udc65 =
\ud835\udc652\u22121
\ud835\udc652+1
; \ud835\udc65 = \u22121.
\ud835\udc4e) \u2203 \ud835\udc53 \u22121 = 0;
\ud835\udc4f) lim
\ud835\udc65\u2192\u22121
\ud835\udc652 \u2212 1
\ud835\udc652 + 1
= 0, \ud835\udc5d\ud835\udc5c\ud835\udc5f\ud835\udc61\ud835\udc4e\ud835\udc5b\ud835\udc61\ud835\udc5c \u2203 lim
\ud835\udc65\u2192\u22121
\ud835\udc53 \ud835\udc65 ;
c) lim
\ud835\udc65\u2192\u22121
\ud835\udc652\u22121
\ud835\udc652+1
= \ud835\udc53 \u22121 .
Logo a função é contínua em \ud835\udc65 = \u22121.
CONTINUIDADE
Exemplos
3) f x = \u124a
\ud835\udc65 + 1, \ud835\udc65 < 1
2 \u2212 \ud835\udc65, \ud835\udc65 \u2265 1
; \ud835\udc65 = 1.
\ud835\udc4e) \u2203\ud835\udc53 1 = 2 \u2212 1 \u2212 1;
\ud835\udc4f) lim
\ud835\udc65\u21921
\ud835\udc53 \ud835\udc65 = \u1250
lim
\ud835\udc65\u21921\u2212
\ud835\udc65 + 1 = 2
lim
\ud835\udc65\u21921+
2 \u2212 \ud835\udc65 = 1
, \ud835\udc5d\ud835\udc5c\ud835\udc5f\ud835\udc61\ud835\udc4e\ud835\udc5b\ud835\udc61\ud835\udc5c \u2204 lim
\ud835\udc65\u21921
\ud835\udc53(\ud835\udc65) .
OBS.: Se uma função NÃO é contínua em um ponto \ud835\udc4e, 
dizemos que ela é descontínua neste ponto;
Logo a função não é
contínua em \ud835\udc65 = 1;
CONTINUIDADE 
Os seguintes tipos de funções são contínuos em cada ponto de seus
domínios:
\u2022 Funções Polinomiais; 
\u2022 Funções Racionais;
\u2022 Função Raiz (\ud835\udc66 = \ud835\udc5b \ud835\udc65, \ud835\udc5b \ud835\udc62\ud835\udc5a \ud835\udc5bú\ud835\udc5a\ud835\udc52\ud835\udc5f\ud835\udc5c \ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc52\ud835\udc56\ud835\udc5f\ud835\udc5c \ud835\udc5d\ud835\udc5c\ud835\udc60\ud835\udc56\ud835\udc61\ud835\udc56\ud835\udc63\ud835\udc5c \ud835\udc5a\ud835\udc4e\ud835\udc56\ud835\udc5c\ud835\udc5f \ud835\udc5e 1);
\u2022 Funções Trigonométricas;
\u2022 Funções Trigonométricas inversas;
\u2022 Funções Exponenciais;
\u2022 Funções Logarítmicas.
CONTINUIDADE 
Alguns Exemplos:
CONTINUIDADE - Propriedades
Se as funções \ud835\udc53 e \ud835\udc54 são contínuas em \ud835\udc65 = \ud835\udc50, então as seguintes
combinações são contínuas em \ud835\udc65 = \ud835\udc50.
1. SOMAS: \ud835\udc53 + \ud835\udc54
2. DIFERENÇAS: \ud835\udc53 \u2212 \ud835\udc54
3. PRODUTOS: \ud835\udc53. \ud835\udc54
4. CONSTANTES MÚLTIPLAS: \ud835\udc58. \ud835\udc53, \ud835\udc5d\ud835\udc4e\ud835\udc5f\ud835\udc4e \ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59\ud835\udc5e\ud835\udc62\ud835\udc52\ud835\udc5f \ud835\udc5bú\ud835\udc5a\ud835\udc52\ud835\udc5f\ud835\udc5c \ud835\udc58
5. QUOCIENTES: \ud835\udc53
\ud835\udc54
, \ud835\udc62\ud835\udc5a\ud835\udc4e \ud835\udc63\ud835\udc52\ud835\udc67 \ud835\udc5e\ud835\udc62\ud835\udc52 \ud835\udc54(\ud835\udc50) \u2260 0
Exercícios
\u2713 Verifique se a função dada é contínua no ponto
indicado:
1) \ud835\udc53 \ud835\udc65 =
\u22123\ud835\udc652\u2212\ud835\udc65+1
\ud835\udc653\u22121
; \ud835\udc65 = \u22121.
2) \ud835\udc54 \ud835\udc65 = \u1244
\u22121 \ud835\udc60\ud835\udc52 \ud835\udc65 < 0
1 \ud835\udc60\ud835\udc52 \ud835\udc65 \u2265 0
; \ud835\udc65 = 0.
3) \u210e \ud835\udc65 = \u124a
\ud835\udc652\u22124
\ud835\udc65\u22122
, \ud835\udc60\ud835\udc52 \ud835\udc65 \u2260 2
3 \ud835\udc60\ud835\udc52 \ud835\udc65 = 2
; \ud835\udc65 = 2.
4) \ud835\udc57 \ud835\udc65 =
|\ud835\udc65+1|
\ud835\udc652\u22124
; \ud835\udc65 = \u22121.
5) k \ud835\udc65 = \u1250
\ud835\udc652 \u2212 1 \ud835\udc60\ud835\udc52 \ud835\udc65 < \u22122
\u22123\ud835\udc65
2
\ud835\udc60\ud835\udc52 \ud835\udc65 \u2265 \u22122
; \ud835\udc65 = \u22122.
6) l \ud835\udc65 = \u124a
\ud835\udc652\u221216
8\u22122\ud835\udc65
\ud835\udc60\ud835\udc52 \ud835\udc65 \u2260 4
2\ud835\udc65 \u2212 4 \ud835\udc60\ud835\udc52 \ud835\udc65 = 4
; \ud835\udc65 = 4.
7) m \ud835\udc65 =
1\u2212\ud835\udc652
\ud835\udc65\u22121
\ud835\udc60\ud835\udc52 \ud835\udc65 > 1
2\ud835\udc652\u22122
1\u2212\ud835\udc65
\ud835\udc60\ud835\udc52 \ud835\udc65 < 1
1 \u2212 5\ud835\udc65 \ud835\udc60\ud835\udc52 \ud835\udc65 = 1
; \ud835\udc65 = 1.