Limites de Funções em Matemática para Negócios

•

ESTÁCIO

0

Erica Odoriges

03/05/2020

E aí, curtiu este material?

Ajude a incentivar outros estudantes a melhorar o conteúdo

Gostou desse material? Compartilhe! 🧡

Matemática para Negócios

9.659 Materiais compartilhados

Baixe o app para aproveitar ainda mais

Leia os materiais offline, sem usar a internet. Além de vários outros recursos!

Prévia do material em texto

Matemática para Negócios
André Brochi
Aula 9
Conceito intuitivo
O limite de uma função num determinado valor de x, isto é, , é definido como aquele valor que a função assume nas vizinhanças de x0.
Limite de uma função
*
Exemplo 1:
função contínua	 função descontínua
 				 
*
Gráf8
		-1		0		2
		-0.9		0.5		2
		-0.8		1		2
		-0.7		2		2
		-0.6				2
		-0.5
		-0.4
		-0.3
		-0.2
		-0.1
		0
		0.1
		0.2
		0.3
		0.4
		0.5
		0.6
		0.7
		0.8
		0.9
		1
		1.1
		1.2
		1.3
		1.4
		1.5
		1.6
		1.7
		1.8
		1.9
		2
		2.1
		2.2
		2.3
		2.4
		2.5
		2.6
		2.7
		2.8
		2.9
		3
x
-5
4
0
-4.7
4
1
-4.4
4
2
-4.1
4
3
-3.8
4
-3.5
-3.2
-2.9
-2.6
-2.3
-2
-1.7
-1.4
-1.1
-0.8
-0.5
-0.2
0.1
0.4
0.7
1
1.3
1.6
1.9
2.2
2.5
2.8
3.1
3.4
3.7
4
4.3
4.6
4.9
5.2
5.5
5.8
6.1
6.4
6.7
7
Plan1
		x		f		pont hor				pont vert
		-3		-11		0		4		2		0
		-2.9		-10.7		0.5		4		2		1
		-2.8		-10.4		1		4		2		2
		-2.7		-10.1		2		4		2		3
		-2.6		-9.8						2		4
		-2.5		-9.5
		-2.4		-9.2
		-2.3		-8.9
		-2.2		-8.6
		-2.1		-8.3
		-2		-8
		-1.9		-7.7
		-1.8		-7.4
		-1.7		-7.1
		-1.6		-6.8
		-1.5		-6.5
		-1.4		-6.2
		-1.3		-5.9
		-1.2		-5.6
		-1.1		-5.3
		-1		-5
		-0.9		-4.7
		-0.8		-4.4
		-0.7		-4.1
		-0.6		-3.8
		-0.5		-3.5
		-0.4		-3.2
		-0.3		-2.9
		-0.2		-2.6
		-0.1		-2.3
		0		-2
		0.1		-1.7
		0.2		-1.4
		0.3		-1.1
		0.4		-0.8
		0.5		-0.5
		0.6		-0.2
		0.7		0.1
		0.8		0.4
		0.9		0.7
		1		1
		1.1		1.3
		1.2		1.6
		1.3		1.9
		1.4		2.2
		1.5		2.5
		1.6		2.8
		1.7		3.1
		1.8		3.4
		1.9		3.7
		2		4
		2.1		4.3
		2.2		4.6
		2.3		4.9
		2.4		5.2
		2.5		5.5
		2.6		5.8
		2.7		6.1
		2.8		6.4
		2.9		6.7
		3		7
Plan1
		
x
f(x)
Plan2
		x		f		pont hor				pont vert
		-3		0.1111111111		0		4		2		0
		-2.9		0.1189060642		0.5		4		2		1
		-2.8		0.1275510204		1		4		2		2
		-2.7		0.1371742112		2		4		2		3
		-2.6		0.1479289941						2		4
		-2.5		0.16
		-2.4		0.1736111111
		-2.3		0.1890359168
		-2.2		0.2066115702
		-2.1		0.2267573696
		-2		0.25
		-1.9		0.2770083102
		-1.8		0.3086419753
		-1.7		0.3460207612
		-1.6		0.390625
		-1.5		0.4444444444
		-1.4		0.5102040816
		-1.3		0.5917159763
		-1.2		0.6944444444
		-1.1		0.826446281
		-1		1
		-0.9		1.2345679012
		-0.8		1.5625
		-0.7		2.0408163265
		-0.6		2.7777777778
		-0.5		4
		-0.4		6.25
		-0.3		11.1111111111
		-0.2		25
		-0.1		100
		0.1		100
		0.2		25
		0.3		11.1111111111
		0.4		6.25
		0.5		4
		0.6		2.7777777778
		0.7		2.0408163265
		0.8		1.5625
		0.9		1.2345679012
		1		1
		1.1		0.826446281
		1.2		0.6944444444
		1.3		0.5917159763
		1.4		0.5102040816
		1.5		0.4444444444
		1.6		0.390625
		1.7		0.3460207612
		1.8		0.3086419753
		1.9		0.2770083102
		2		0.25
		2.1		0.2267573696
		2.2		0.2066115702
		2.3		0.1890359168
		2.4		0.1736111111
		2.5		0.16
		2.6		0.1479289941
		2.7		0.1371742112
		2.8		0.1275510204
		2.9		0.1189060642
		3		0.1111111111
Plan2
		
x
f(x)
Plan3
		x		f		pont hor				pont vert
		-3		-1		0		4		2		0
		-2.9		-0.9		0.5		4		2		1
		-2.8		-0.8		1		4		2		2
		-2.7		-0.7		2		4		2		3
		-2.6		-0.6						2		4
		-2.5		-0.5
		-2.4		-0.4
		-2.3		-0.3
		-2.2		-0.2
		-2.1		-0.1
		-2		0
		-1.9		0.1
		-1.8		0.2
		-1.7		0.3
		-1.6		0.4
		-1.5		0.5
		-1.4		0.6
		-1.3		0.7
		-1.2		0.8
		-1.1		0.9
		-1		1
		-0.9		1.1
		-0.8		1.2
		-0.7		1.3
		-0.6		1.4
		-0.5		1.5
		-0.4		1.6
		-0.3		1.7
		-0.2		1.8
		-0.1		1.9
		0		2
		0.1		2.1
		0.2		2.2
		0.3		2.3
		0.4		2.4
		0.5		2.5
		0.6		2.6
		0.7		2.7
		0.8		2.8
		0.9		2.9
		1		3
		1.1		3.1
		1.2		3.2
		1.3		3.3
		1.4		3.4
		1.5		3.5
		1.6		3.6
		1.7		3.7
		1.8		3.8
		1.9		3.9
		2		0
		2.1		4.1
		2.2		4.2
		2.3		4.3
		2.4		4.4
		2.5		4.5
		2.6		4.6
		2.7		4.7
		2.8		4.8
		2.9		4.9
		3		5
Plan3
						2
x
f(x)
4
Gráf1
		-3
		-2.9
		-2.8
		-2.7
		-2.6
		-2.5
		-2.4
		-2.3
		-2.2
		-2.1
		-2
		-1.9
		-1.8
		-1.7
		-1.6
		-1.5
		-1.4
		-1.3
		-1.2
		-1.1
		-1
		-0.9
		-0.8
		-0.7
		-0.6
		-0.5
		-0.4
		-0.3
		-0.2
		-0.1
		0.1
		0.2
		0.3
		0.4
		0.5
		0.6
		0.7
		0.8
		0.9
		1
		1.1
		1.2
		1.3
		1.4
		1.5
		1.6
		1.7
		1.8
		1.9
		2
		2.1
		2.2
		2.3
		2.4
		2.5
		2.6
		2.7
		2.8
		2.9
		3
x
0.1111111111
0.1189060642
0.1275510204
0.1371742112
0.1479289941
0.16
0.1736111111
0.1890359168
0.2066115702
0.2267573696
0.25
0.2770083102
0.3086419753
0.3460207612
0.390625
0.4444444444
0.5102040816
0.5917159763
0.6944444444
0.826446281
1
1.2345679012
1.5625
2.0408163265
2.7777777778
4
6.25
11.1111111111
25
100
100
25
11.1111111111
6.25
4
2.7777777778
2.0408163265
1.5625
1.2345679012
1
0.826446281
0.6944444444
0.5917159763
0.5102040816
0.4444444444
0.390625
0.3460207612
0.3086419753
0.2770083102
0.25
0.2267573696
0.2066115702
0.1890359168
0.1736111111
0.16
0.1479289941
0.1371742112
0.1275510204
0.1189060642
0.1111111111
Plan1
		x		f		pont hor				pont vert
		-3		-11		0		4		2		0
		-2.9		-10.7		0.5		4		2		1
		-2.8		-10.4		1		4		2		2
		-2.7		-10.1		2		4		2		3
		-2.6		-9.8						2		4
		-2.5		-9.5
		-2.4		-9.2
		-2.3		-8.9
		-2.2		-8.6
		-2.1		-8.3
		-2		-8
		-1.9		-7.7
		-1.8		-7.4
		-1.7		-7.1
		-1.6		-6.8
		-1.5		-6.5
		-1.4		-6.2
		-1.3		-5.9
		-1.2		-5.6
		-1.1		-5.3
		-1		-5
		-0.9		-4.7
		-0.8		-4.4
		-0.7		-4.1
		-0.6		-3.8
		-0.5		-3.5
		-0.4		-3.2
		-0.3		-2.9
		-0.2		-2.6
		-0.1		-2.3
		0		-2
		0.1		-1.7
		0.2		-1.4
		0.3		-1.1
		0.4		-0.8
		0.5		-0.5
		0.6		-0.2
		0.7		0.1
		0.8		0.4
		0.9		0.7
		1		1
		1.1		1.3
		1.2		1.6
		1.3		1.9
		1.4		2.2
		1.5		2.5
		1.6		2.8
		1.7		3.1
		1.8		3.4
		1.9		3.7
		2		4
		2.1		4.3
		2.2		4.6
		2.3		4.9
		2.4		5.2
		2.5		5.5
		2.6		5.8
		2.7		6.1
		2.8		6.4
		2.9		6.7
		3		7
Plan1
		
x
f(x)
Plan2
		x		f		pont hor				pont vert
		-3		0.1111111111		0		4		2		0
		-2.9		0.1189060642		0.5		4		2		1
		-2.8		0.1275510204		1		4		2		2
		-2.7		0.1371742112		2		4		2		3
		-2.6		0.1479289941						2		4
		-2.5		0.16
		-2.4		0.1736111111
		-2.3		0.1890359168
		-2.2		0.2066115702
		-2.1		0.2267573696
		-2		0.25
		-1.9		0.2770083102
		-1.8		0.3086419753
		-1.7		0.3460207612
		-1.6		0.390625
		-1.5		0.4444444444
		-1.4		0.5102040816
		-1.3		0.5917159763
		-1.2		0.6944444444
		-1.1		0.826446281
		-1		1
		-0.9		1.2345679012
		-0.8		1.5625
		-0.7		2.0408163265
		-0.6		2.7777777778
		-0.5		4
		-0.4		6.25
		-0.3		11.1111111111
		-0.2		25
		-0.1		100
		0.1		100
		0.2		25
		0.3		11.1111111111
		0.4		6.25
		0.5		4
		0.6		2.7777777778
		0.7		2.0408163265
		0.8		1.5625
		0.9		1.2345679012
		1		1
		1.1		0.826446281
		1.2		0.6944444444
		1.3		0.5917159763
		1.4		0.5102040816
		1.5		0.4444444444
		1.6		0.390625
		1.7		0.3460207612
		1.8		0.3086419753
		1.9		0.2770083102
		2		0.25
		2.1		0.2267573696
		2.2		0.2066115702
		2.3		0.1890359168
		2.4		0.1736111111
		2.5		0.16
		2.6		0.1479289941
		2.7		0.1371742112
		2.8		0.1275510204
		2.9		0.1189060642
		3		0.1111111111
Plan2
		
x
f(x)
Plan3
		x		f		pont hor				pont vert
		-3		-1		0		4		2		0
		-2.9		-0.9		0.5		4		2		1
		-2.8		-0.8		1		4		2		2
		-2.7		-0.7		2		4		2		3
		-2.6		-0.6						2		4
		-2.5		-0.5
		-2.4		-0.4
		-2.3		-0.3
		-2.2		-0.2
		-2.1		-0.1
		-2		0
		-1.9		0.1
		-1.8		0.2
		-1.7		0.3
		-1.6		0.4
		-1.5		0.5
		-1.4		0.6
		-1.3		0.7
		-1.2		0.8
		-1.1		0.9
		-1		1
		-0.9		1.1
		-0.8		1.2
		-0.7		1.3
		-0.6		1.4
		-0.5		1.5
		-0.4		1.6
		-0.3		1.7
		-0.2		1.8
		-0.1		1.9
		0		2
		0.1		2.1
		0.2		2.2
		0.3		2.3
		0.4		2.4
		0.5		2.5
		0.6		2.6
		0.7		2.7
		0.8		2.8
		0.9		2.9
		1		3
		1.1		3.1
		1.2		3.2
		1.3		3.3
		1.4		3.4
		1.5		3.5
		1.6		3.6
		1.7		3.7
		1.8		3.8
		1.9		3.9
		2		0
		2.1		4.1
		2.2		4.2
		2.3		4.3
		2.4		4.4
		2.5		4.5
		2.6		4.6
		2.7		4.7
		2.8		4.8
		2.94.9
		3		5
Plan3
						2
x
f(x)
4
Limites de funções contínuas
Se uma função f(x) é contínua em , então:
O valor do limite da função para x tendendo a x0 é igual ao valor da função quando x é igual a x0.
*
Exemplo 2: encontrar
 
*
	X	Y = f(x)
	-1	- 4
	0	-1
	1	2
	1,5	3,5
	1,6	3,8
	1,9	4,7
	1,999	4,997
	1,99999	4,99997
	X	Y = f(x)
	4	11
	3	8
	2,5	6,5
	2,2	5,6
	2,1	5,3
	2,01	5,03
	2,001	5,003
	2,00001	5,00003
Exemplo 3:
a)
b)
c)
*
d)
e) 
*
Gráf9
		-1		2.1		2		0		2
		-0.9		2.2				0.5		2
		-0.8		2.3				1		2
		-0.7		2.4				2		2
		-0.6		2.5						2
		-0.5		2.6
		-0.4		2.7
		-0.3		2.8
		-0.2		2.9
		-0.1		3
		0
		0.1
		0.2
		0.3
		0.4
		0.5
		0.6
		0.7
		0.8
		0.9
		1
		1.1
		1.2
		1.3
		1.4
		1.5
		1.6
		1.7
		1.8
		1.9
x
f(x)
1
4.1
4
4
0
1.1
4.2
4
1
1.2
4.3
4
2
1.3
4.4
4
3
1.4
4.5
4
1.5
4.6
1.6
4.7
1.7
4.8
1.8
4.9
1.9
5
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
Plan1
		x		f		pont hor				pont vert
		-3		-11		0		4		2		0
		-2.9		-10.7		0.5		4		2		1
		-2.8		-10.4		1		4		2		2
		-2.7		-10.1		2		4		2		3
		-2.6		-9.8						2		4
		-2.5		-9.5
		-2.4		-9.2
		-2.3		-8.9
		-2.2		-8.6
		-2.1		-8.3
		-2		-8
		-1.9		-7.7
		-1.8		-7.4
		-1.7		-7.1
		-1.6		-6.8
		-1.5		-6.5
		-1.4		-6.2
		-1.3		-5.9
		-1.2		-5.6
		-1.1		-5.3
		-1		-5
		-0.9		-4.7
		-0.8		-4.4
		-0.7		-4.1
		-0.6		-3.8
		-0.5		-3.5
		-0.4		-3.2
		-0.3		-2.9
		-0.2		-2.6
		-0.1		-2.3
		0		-2
		0.1		-1.7
		0.2		-1.4
		0.3		-1.1
		0.4		-0.8
		0.5		-0.5
		0.6		-0.2
		0.7		0.1
		0.8		0.4
		0.9		0.7
		1		1
		1.1		1.3
		1.2		1.6
		1.3		1.9
		1.4		2.2
		1.5		2.5
		1.6		2.8
		1.7		3.1
		1.8		3.4
		1.9		3.7
		2		4
		2.1		4.3
		2.2		4.6
		2.3		4.9
		2.4		5.2
		2.5		5.5
		2.6		5.8
		2.7		6.1
		2.8		6.4
		2.9		6.7
		3		7
Plan1
		
x
f(x)
Plan2
		x		f		pont hor				pont vert
		-3		0.1111111111		0		4		2		0
		-2.9		0.1189060642		0.5		4		2		1
		-2.8		0.1275510204		1		4		2		2
		-2.7		0.1371742112		2		4		2		3
		-2.6		0.1479289941						2		4
		-2.5		0.16
		-2.4		0.1736111111
		-2.3		0.1890359168
		-2.2		0.2066115702
		-2.1		0.2267573696
		-2		0.25
		-1.9		0.2770083102
		-1.8		0.3086419753
		-1.7		0.3460207612
		-1.6		0.390625
		-1.5		0.4444444444
		-1.4		0.5102040816
		-1.3		0.5917159763
		-1.2		0.6944444444
		-1.1		0.826446281
		-1		1
		-0.9		1.2345679012
		-0.8		1.5625
		-0.7		2.0408163265
		-0.6		2.7777777778
		-0.5		4
		-0.4		6.25
		-0.3		11.1111111111
		-0.2		25
		-0.1		100
		0.1		100
		0.2		25
		0.3		11.1111111111
		0.4		6.25
		0.5		4
		0.6		2.7777777778
		0.7		2.0408163265
		0.8		1.5625
		0.9		1.2345679012
		1		1
		1.1		0.826446281
		1.2		0.6944444444
		1.3		0.5917159763
		1.4		0.5102040816
		1.5		0.4444444444
		1.6		0.390625
		1.7		0.3460207612
		1.8		0.3086419753
		1.9		0.2770083102
		2		0.25
		2.1		0.2267573696
		2.2		0.2066115702
		2.3		0.1890359168
		2.4		0.1736111111
		2.5		0.16
		2.6		0.1479289941
		2.7		0.1371742112
		2.8		0.1275510204
		2.9		0.1189060642
		3		0.1111111111
Plan2
		
x
f(x)
Plan3
		x		f		pont hor				pont vert
		-3		-1		0		4		2		0
		-2.9		-0.9		0.5		4		2		1
		-2.8		-0.8		1		4		2		2
		-2.7		-0.7		2		4		2		3
		-2.6		-0.6						2		4
		-2.5		-0.5
		-2.4		-0.4
		-2.3		-0.3
		-2.2		-0.2
		-2.1		-0.1
		-2		0
		-1.9		0.1
		-1.8		0.2
		-1.7		0.3
		-1.6		0.4
		-1.5		0.5
		-1.4		0.6
		-1.3		0.7
		-1.2		0.8
		-1.1		0.9
		-1		1
		-0.9		1.1
		-0.8		1.2
		-0.7		1.3
		-0.6		1.4
		-0.5		1.5
		-0.4		1.6
		-0.3		1.7
		-0.2		1.8
		-0.1		1.9
		0		2
		0.1		2.1
		0.2		2.2
		0.3		2.3
		0.4		2.4
		0.5		2.5
		0.6		2.6
		0.7		2.7
		0.8		2.8
		0.9		2.9
		1		3
		1.1		3.1
		1.2		3.2
		1.3		3.3
		1.4		3.4
		1.5		3.5
		1.6		3.6
		1.7		3.7
		1.8		3.8
		1.9		3.9
		2		0
		2.1		4.1
		2.2		4.2
		2.3		4.3
		2.4		4.4
		2.5		4.5
		2.6		4.6
		2.7		4.7
		2.8		4.8
		2.9		4.9
		3		5
Plan3
						2
x
f(x)
4
Limites de funções 
descontínuas
Se uma função f(x) é descontínua em x = x0, então para determinar , devemos calcular os valores de f(x) para x se aproximando de x0 (tanto pela direita quanto pela esquerda). 
*
Exemplo 4: encontrar 
 Pela esquerda	 Pela direita
 
*
	x	 f(x)
	-1	1
	0	2
	1	3
	1,5	3,5
	1,6	3,6
	1,9	3,9
	1,999	3,999
	1,99999	3,99999
	x	f(x)
	4	6
	3	5
	2,5	4,5
	2,2	4,2
	2,1	4,1
	2,01	4,01
	2,001	4,001
	2,00001	4,00001
 existe e é igual a 4. 
*
Gráf9
		-1		2.1		2		0		2
		-0.9		2.2				0.5		2
		-0.8		2.3				1		2
		-0.7		2.4				2		2
		-0.6		2.5						2
		-0.5		2.6
		-0.4		2.7
		-0.3		2.8
		-0.2		2.9
		-0.1		3
		0
		0.1
		0.2
		0.3
		0.4
		0.5
		0.6
		0.7
		0.8
		0.9
		1
		1.1
		1.2
		1.3
		1.4
		1.5
		1.6
		1.7
		1.8
		1.9
x
f(x)
1
4.1
4
4
0
1.1
4.2
4
1
1.2
4.3
4
2
1.3
4.4
4
3
1.4
4.5
4
1.5
4.6
1.6
4.7
1.7
4.8
1.8
4.9
1.9
5
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
Plan1
		x		f		pont hor				pont vert
		-3		-11		0		4		2		0
		-2.9		-10.7		0.5		4		2		1
		-2.8		-10.4		1		4		2		2
		-2.7		-10.1		2		4		2		3
		-2.6		-9.8						2		4
		-2.5		-9.5
		-2.4		-9.2
		-2.3		-8.9
		-2.2		-8.6
		-2.1		-8.3
		-2		-8
		-1.9		-7.7
		-1.8		-7.4
		-1.7		-7.1
		-1.6		-6.8
		-1.5		-6.5
		-1.4		-6.2
		-1.3		-5.9
		-1.2		-5.6
		-1.1		-5.3
		-1		-5
		-0.9		-4.7
		-0.8		-4.4
		-0.7		-4.1
		-0.6		-3.8
		-0.5		-3.5
		-0.4		-3.2
		-0.3		-2.9
		-0.2		-2.6
		-0.1		-2.3
		0		-2
		0.1		-1.7
		0.2		-1.4
		0.3		-1.1
		0.4		-0.8
		0.5		-0.5
		0.6		-0.2
		0.7		0.1
		0.8		0.4
		0.9		0.7
		1		1
		1.1		1.3
		1.2		1.6
		1.3		1.9
		1.4		2.2
		1.5		2.5
		1.6		2.8
		1.7		3.1
		1.8		3.4
		1.9		3.7
		2		4
		2.1		4.3
		2.2		4.6
		2.3		4.9
		2.4		5.2
		2.5		5.5
		2.6		5.8
		2.7		6.1
		2.8		6.4
		2.9		6.7
		3		7
Plan1
		
x
f(x)
Plan2
		x		f		pont hor				pont vert
		-3		0.1111111111		0		4		2		0
		-2.9		0.1189060642		0.5		4		2		1
		-2.8		0.1275510204		1		4		2		2
		-2.7		0.1371742112		2		4		2		3
		-2.6		0.1479289941						2		4
		-2.5		0.16
		-2.4		0.1736111111
		-2.3		0.1890359168
		-2.2		0.2066115702
		-2.1		0.2267573696
		-2		0.25
		-1.9		0.2770083102
		-1.8		0.3086419753
		-1.7		0.3460207612
		-1.6		0.390625
		-1.5		0.4444444444
		-1.4		0.5102040816
		-1.3		0.5917159763
		-1.2		0.6944444444
		-1.1		0.826446281
		-1		1
		-0.9		1.2345679012
		-0.8		1.5625
		-0.7		2.0408163265
		-0.6		2.7777777778
		-0.5		4
		-0.4		6.25
		-0.3		11.1111111111
		-0.2		25
		-0.1		100
		0.1		100
		0.2		25
		0.3		11.1111111111
		0.4		6.25
		0.5		4
		0.6		2.7777777778
		0.7		2.0408163265
		0.8		1.5625
		0.9		1.2345679012
		1		1
		1.1		0.826446281
		1.2		0.6944444444
		1.3		0.5917159763
		1.4		0.5102040816
		1.5		0.4444444444
		1.6		0.390625
		1.7		0.3460207612
		1.8		0.3086419753
		1.9		0.2770083102
		2		0.25
		2.1		0.2267573696
		2.2		0.2066115702
		2.3		0.1890359168
		2.4		0.1736111111
		2.5		0.16
		2.6		0.1479289941
		2.7		0.1371742112
		2.8		0.1275510204
		2.9		0.1189060642
		3		0.1111111111
Plan2
		
x
f(x)
Plan3
		x		f		pont hor				pont vert
		-3		-1		0		4		2		0
		-2.9		-0.9		0.5		4		2		1
		-2.8		-0.8		1		4		2		2
		-2.7		-0.7		2		4		2		3
		-2.6		-0.6						2		4
		-2.5		-0.5
		-2.4		-0.4
		-2.3		-0.3
		-2.2		-0.2
		-2.1		-0.1
		-2		0
		-1.9		0.1
		-1.8		0.2
		-1.7		0.3
		-1.6		0.4
		-1.5		0.5
		-1.4		0.6
		-1.3		0.7
		-1.2		0.8
		-1.1		0.9
		-1		1
		-0.9		1.1
		-0.8		1.2
		-0.7		1.3
		-0.6		1.4
		-0.5		1.5
		-0.4		1.6
		-0.3		1.7
		-0.2		1.8
		-0.1		1.9
		0		2
		0.1		2.1
		0.2		2.2
		0.3		2.3
		0.4		2.4
		0.5		2.5
		0.6		2.6
		0.7		2.7
		0.8		2.8
		0.9		2.9
		1		3
		1.1		3.1
		1.2		3.2
		1.33.3
		1.4		3.4
		1.5		3.5
		1.6		3.6
		1.7		3.7
		1.8		3.8
		1.9		3.9
		2		0
		2.1		4.1
		2.2		4.2
		2.3		4.3
		2.4		4.4
		2.5		4.5
		2.6		4.6
		2.7		4.7
		2.8		4.8
		2.9		4.9
		3		5
Plan3
						2
x
f(x)
4
 Como calcular algebricamente?
*
Matemática para Negócios
André Brochi
Atividade 9
Calcule:
*
Atividade
*
(
)
x
f
x
x
0
lim
®
-2-10123456-10123
x
(
)
2
3
-
=
x
x
f
(
)
2
1
x
x
f
=
0
x
x
=
)
(
)
(
lim
0
0
x
f
x
f
x
x
=
®
(
)
:
1
3
lim
2
-
®
x
x
(
)
9
7
9
9
3
1
1
3
1
9
1
1
3
1
3
1
1
lim
2
2
3
1
=
+
-
=
=
+
-
=
+
-
÷
ø
ö
ç
è
æ
=
+
-
®
x
x
x
(
)
(
)
8
3
5
3
5
3
lim
5
=
+
=
+
-
-
=
+
-
-
®
x
x
(
)
4
4
lim
3
=
®
x
-10123456-10123
f(x)x
2
2
4
2
0
4
0
2
4
lim
2
2
0
=
-
-
=
-
-
=
÷
÷
ø
ö
ç
ç
è
æ
-
-
®
x
x
x
?
2
4
lim
2
2
=
÷
÷
ø
ö
ç
ç
è
æ
-
-
®
x
x
x
)
(
lim
0
x
f
x
x
®
:
2
4
lim
2
2
÷
÷
ø
ö
ç
ç
è
æ
-
-
®
x
x
x
-10123456-10123
f(x)x
(
)
2
4
2
-
-
=
x
x
x
f
÷
÷
ø
ö
ç
ç
è
æ
-
-
®
2
4
lim
2
2
x
x
x
=
÷
÷
ø
ö
ç
ç
è
æ
-
®
x
x
x
x
3
lim
 
a)
2
5
=
÷
÷
ø
ö
ç
ç
è
æ
-
®
x
x
x
x
3
lim
 
b)
2
0