Aula_09 Matematica para negocios

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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
		-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
x
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
						4
x
f(x)
Gráf1
		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
x
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
						4
x
f(x)
*
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
 
*
*
Exemplo 3:
a)
b)
c)
*
*
d)
e) 
*
Gráf9
		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
x
f(x)
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
						4
x
f(x)
*
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). 
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Exemplo 4: encontrar 
 Pela esquerda	 Pela direita
 
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*
 existe e é igual a 4. 
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Gráf9
		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
x
f(x)
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)
Plan3x		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
						4
x
f(x)
*
 Como calcular algebricamente?
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Matemática para Negócios
André Brochi
Atividade 9
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Calcule:
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Atividade
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