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de Estudo de C\303\241lculo\302\251 Maplesoft, a division of Waterloo Maple Inc., 2005Cap\303\255tulo1:Interpreta\303\247\303\243o do Conceito de Limite
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Introdu\303\247\303\243o</Font></Text-field>A palavra "limite" \303\251 usada em uma variedade de circunst\303\242ncias cotidianas. Por exemplo, todos n\303\263s j\303\241 ouvimos um pai ou m\303\243e dizer "Estou chegando ao limite da minha paci\303\252ncia com voc\303\252!" A defini\303\247\303\243o matem\303\241tica intuitiva \303\251 muito parecida. O "limite" \303\251 determinado pelo que acontece nas "proximidades".Portanto, n\303\263s dizemos que L \303\251 o limite da fun\303\247\303\243o f(x) quando x se aproxima de um n\303\272mero a, se os valores de f(x) (para x "pr\303\263ximo" de a ) s\303\243o "pr\303\263ximos" de L. Entretanto, o valor de L pode ser bem diferente do n\303\272mero f(a), caso este n\303\272mero exista !Este cap\303\255tulo desenvolve uma interpreta\303\247\303\243o intuitiva do conceito de limite. Primeiro, vamos encontrar limites( limits) de um gr\303\241fico (graph) ; depois desenvolver algumas t\303\251cnicas b\303\241sicas para a avalia\303\247\303\243o anal\303\255lica de limites. Essas t\303\251cnicas ser\303\243o combinadas para desenvolver o conceito de limite lateral( one-sided ) e dar base para uma vis\303\243o inicial a dois limites trigonom\303\251tricos( trigonometric) especiais. O cap\303\255tulo termina chamando \303\240 aten\303\247\303\243o para como gr\303\241ficos( graphs) e tabela de valores( values) podem levar a erro.
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Gloss\303\241rio </Font></Text-field>Os seguintes termos, neste cap\303\255tulo, t\303\252m conex\303\243o com o Dicion\303\241rio de Matem\303\241tica Maple ( Maple Math Dictionary).
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Inicializa\303\247\303\243o</Font></Text-field>restart;
with( plots ):
<Text-field style="Heading 1" layout="Heading 2" spaceabove="7"><Font encoding="UTF-8">Avalia\303\247\303\243o Gr\303\241fica de Limites </Font></Text-field>O primeiro passo para compreender limites \303\251 ser capaz de identific\303\241-los em um gr\303\241fico de uma fun\303\247\303\243o y = f(x). Uma vez dominado isso, torna-se necess\303\241rio determinar limites da descri\303\247\303\243o alg\303\251brica da fun\303\247\303\243o. Isso \303\251 mais simples quando a fun\303\247\303\243o \303\251 definida no ponto limite e n\303\243o existem "problemas" no ponto de limite.
Deve-se estar consciente de que este m\303\251todo n\303\243o \303\251 infal\303\255vel. Os "problemas" mais comuns s\303\243o os que surgem de divis\303\243o por zero e de raiz de \303\255ndice par ( even ) de n\303\272mero negativo.
Esse conceito \303\251 ilustrado no Exemplo 1.1.
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Exemplo 1.1: <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> </Text-field>Considere a fun\303\247\303\243of := x -> ( x^4 - 13*x^3 + 62*x^2 - 120*x + 64 ) / ( x - 4 ):
'f'(x) = f(x);Antes da avalia\303\247\303\243o de qualquer limite, \303\251 aconselh\303\241vel fazer o gr\303\241fico da fun\303\247\303\243o. No gr\303\241fico da Figura 1.1, nota-se que a fun\303\247\303\243o n\303\243o \303\251 definida para x = 4.
Figura1.1 Gr\303\241fico de 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 no intervalo [0, 6]. A fun\303\247\303\243o n\303\243o \303\251 definida em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkktRjY2LVEiPUYnRjJGOUY7Rj1GP0ZBRkNGRS9GSFEsMC4yNzc3Nzc4ZW1GJy9GS0ZQRjUtSSNtbkdGJDYkUSI0RidGMkYy.
A primeira tentativa quando se avalia um limite \303\251 tentar avaliar ( evaluate ) a fun\303\247\303\243o no ponto de limite. Neste caso, a fun\303\247\303\243o \303\251 definida no ponto x = 1 (veja a cruz azul no gr\303\241fico acina) e o valor da fun\303\247\303\243o no ponto x = 1 \303\251f(1);Com base nesta computa\303\247\303\243o e no gr\303\241fico, isto aparece comoLimit( f(x), x=1 ) = f(1);Para confirmar o resultado, note que Maple informa o valor deste limite como sendoLimit( f(x), x=1 ) = limit( f(x), x=1 );A mesma t\303\251cnica poderia ser aplicada para qualquer outros pontos de limites--- exceto x = 4. No ponto x = 4 esta fun\303\247\303\243o n\303\243o \303\251 definida. Pelo gr\303\241fico, est\303\241 claro que todos os valores da fun\303\247\303\243o para x perto de x = 4, se aproximam do mesmo valor, a saber 5. Uma deriva\303\247\303\243o anal\303\255tica para este resultado ser\303\241 desenvolvida a seguirxxxxxxxxxxxxxxxxxxxxxxxxxxxQuest\303\243o 1.1Encontre o limite: 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
<Text-field style="Heading 1" layout="Heading 2" spaceabove="7"><Font encoding="UTF-8">M\303\251todos Anal\303\255ticos B\303\241sicos</Font></Text-field>O "problema" t\303\255pico que impede uma avalia\303\247\303\243o direta da fun\303\247\303\243o no ponto limite \303\251 "divis\303\243o por zero".
Se a fun\303\247\303\243o \303\251 uma fun\303\247\303\243o racional ( rational ) e o numerador( numerator ) \303\251 tambem zero no ponto limite, digamos LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRImFGJ0Y0RjdGPkYrRj4= , ent\303\243o (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEoJm1pbnVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRImFGJ0Y0RjdGPkYrRj4= ) deve ser um fator ( factor ) comum de ambos numerador ( numerator) e denominador. Depois do cancelamento ( cancellation ) de todos os fatores comuns, pode-se tentar de novo a avalia\303\247\303\243o direta . Isto \303\251 ilustrado no exemplo 1.2
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Exemplo 1.2: <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> </Text-field>Considere a fun\303\247\303\243of := x -> ( x^4 - 13*x^3 + 62*x^2 - 120*x + 64 ) / ( x - 4 ):
'f'(x) = f(x);Cujo gr\303\241fico aparece na figura 1.2.
Figura 1.2 Gr\303\241fico de 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 no intervalo [0, 6]. A fun\303\247\303\243o n\303\243o \303\251 definida em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkktRjY2LVEiPUYnRjJGOUY7Rj1GP0ZBRkNGRS9GSFEsMC4yNzc3Nzc4ZW1GJy9GS0ZQRjUtSSNtbkdGJDYkUSI0RidGMkYy.
A primeira tentativa quando se avalia um limite \303\251 avaliar( evaluate ) a fun\303\247\303\243o no ponto de limite. Neste caso a fun\303\247\303\243o n\303\243o \303\251 definida em x = 4, como se v\303\252 def(4);Ap\303\263s uma inspe\303\247\303\243o mais detalhada v\303\252-se que ambos, numerador e denominador ( denominator ) s\303\243o zero quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiNEYnRj5GPkYrRj4=, Isso \303\251eval( numer(f(x)), x=4 );
eval( denom(f(x)), x=4 );Isto significa que o numerador contem o fator ( factor ) (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEoJm1pbnVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiNEYnRj5GPkYrRj4=), como vemos defactor(numer(f(x)));Se este fator comum \303\251 "cancelado", temos a fun\303\247\303\243og := unapply(factor(f(x)),x):
'g'(x) = g(x);A nova fun\303\247\303\243o, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJnRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RidGNEY3Rj5GPkYrRj5GK0Y+, \303\251 id\303\252ntica a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RidGNEY3Rj5GPkYrRj5GK0Y+ exceto que ela \303\251 definida em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiNEYnRj5GPkYrRj4=. Isto \303\251 ilustrado na Figura 1.3.
Figura 1.3 Gr\303\241fico de 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 (em negro), e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYnUSJnRicvJSdmYW1pbHlHUSZzZXJpZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GPlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkgvJSlzdHJldGNoeUdGSC8lKnN5bW1ldHJpY0dGSC8lKGxhcmdlb3BHRkgvJS5tb3ZhYmxlbGltaXRzR0ZILyUnYWNjZW50R0ZILyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGVy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNidRInhGJ0Y0RjdGOkY9RkRGREZERitGRA== (em verde), por Exemplo 1.2.
O objetivo do Exemplo !.2 \303\251 ilustrar que mudando a defini\303\247\303\243o da fun\303\247\303\243o em um ponto n\303\243o muda o valor do limite, ou mais formalmente, queLimit( f(x), x=4 ) = Limit( g(x), x=4 );
O limite pode ser avaliado como no exemplo anteriorLimit( g(x), x=4 ) = g(4);
Este resultado parece ser consistente com a informa\303\247\303\243o gr\303\241fica. Mais ainda, Maple informa que o valor do limite original \303\251Limit( f(x), x=4 ) = limit( f(x), x=4 );
Note que se o numerador n\303\243o for zero no ponto limite ent\303\243o nenhum cancelamento ( cancellation ) \303\251 poss\303\255vel. Esta situa\303\247\303\243o ser\303\241 resolvida em detalhe no Cap\303\255tulo 6, o Cap\303\255tulo Limites Infinitos.XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX QUESTAO 1.2Quest\303\243o 1.2Encontre o limite: 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
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Fun\303\247\303\265es Definidas por Partes e Limites Laterais \303\240 Esquerda e \303\240 Direita </Font></Text-field>Fun\303\247\303\265es Definidas por Partes (Piecewise-defined) apresentam uma nova dificuldade ( wrinkle) al\303\251m da encontrada com fun\303\247\303\265es racionais (rational ). A dificuldade com uma fun\303\247\303\243o piecewise-defined aparece quando o ponto limite \303\251 tamb\303\251m um ponto de transi\303\247\303\243o entre duas partes da defini\303\247\303\243o da fun\303\247\303\243o.Nesta situa\303\247\303\243o, geralmente n\303\243o \303\251 poss\303\255vel simplificar algebricamente o problema, tanto que ele pode ser avaliado por substitui\303\247\303\243o direta. Em lugar disso, as duas diferentes dire\303\247\303\265es pelas quais pode-se aproximar do ponto de limite levam a dois limites laterais , \303\240 esquerda e a direita.
Cada um destes limites \303\251 usualmente simples de avaliar j\303\241 que envolve somente uma parte da defini\303\247\303\243o da fun\303\247\303\243o definida por partes ( piecewise-defined ). Se estes dois limites existem e t\303\252m o mesmo valor, ent\303\243o o limite lateral ( two-sided ) existe e tem o mesmo valor que os dois limites \303\240 esquerda e a direita. ( one-sided . )Exemplos 1.3 e 1.4 ilustram estas id\303\251ias.
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Exemplo 1.3: <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> </Text-field>Considere a fun\303\247\303\243of := x -> piecewise( x<1, -x^2+2, x>=1, sqrt(x-1)-2*x/3+2 ):
'f'(x) = f(x);cujo gr\303\241fico aparece na Figura 1.4.
plot( f(x), x=0..6, y=0..3, discont=true, view=[0..6,0..3] );Figura 1.4 Exemplo 1.3: uma fun\303\247\303\243o piecewise-defined que salta em 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.
A Figura 1.4, convincentemente, mostra que o limite lateral desta fun\303\247\303\243o n\303\243o existe em x = 1. Isto \303\251 verdade, mesmo que n\303\243o exista o problema de avalia\303\247\303\243o direta da fun\303\247\303\243o no ponto limite, como \303\251 visto def(1);Entretanto, o Maple informa que o limite n\303\243o existe, como \303\251 visto emlimit( f(x), x=1 );Para entender porque este limite n\303\243o existe, considere os dois limites laterais, \303\240 esuqerda e \303\240 direita . O limite pela esquerda( limit from the left )\303\251q1 := Limit( f(x), x=1, left ) = Limit( -x^2+2, x=1 ):
q1;
`` = eval( -x^2+2, x=1 );enquanto que o limite pela direita( limit from the right )\303\251q2 := Limit( f(x), x=1, right ) = Limit( sqrt(x-1)-2*x/3+2, x=1 ):
q2;
`` = eval( sqrt(x-1)-2*x/3+2, x=1 );O limite lateral ( two-sided ) n\303\243o existe porque os limites pela esquerda e pela direita t\303\252m valores diferentes Este resultado \303\251 confirmado pelo Maple via Limit( f(x), x=1 ) = limit(f(x),x=1);
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Exemplo 1.4: <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> </Text-field>Considere a fun\303\247\303\243of := x -> piecewise( x<1, -x^2+2, x>=1, sqrt(x-1)-x+2 ):
'f'(x) = f(x);Esta fun\303\247\303\243o \303\251 muito parecida com a fun\303\247\303\243o do exemplo 1.3. A Figura 1.5 mostra que existe,entretanto, uma diferen\303\247a fundamental.
plot( f(x), x=0..6, y=0..3, discont=true );
Figura 1.5 Exemplo 1.4: Uma fun\303\247\303\243o piecewise-defined sem um salto.
Fugura 1.5 sugere que os valores desta fun\303\247\303\243o em nenhum dos lados de x = 1 se aproximam do mesmo valor-- a saber,1. Para confirmar isto, olhe para os dois limites laterais, pela esquerda e pela direita .O limite pela esquerda ( limit from the left ) \303\251q1 := Limit( f(x), x=1, left ) = Limit( -x^2+2, x=1 ):
q1;
`` = eval( -x^2+2, x=1 );
enquanto, o limite pela direita ( limit from the right ) \303\251q2 := Limit( f(x), x=1, right ) = Limit( sqrt(x-1)-x+2, x=1 ):
q2;
`` = eval( sqrt(x-1)-x+2, x=1 );
Podemos concluir que existe o limite two-sided e tem valor 1 porque os limites one-sided existem e t\303\252m o mesmo valor. Maple confirma esta conclus\303\243o viaLimit( f(x), x=1 ) = limit(f(x),x=1);XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXQUESTAO 1.3Quest\303\243o 1.4Encontre o limite: 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Se o limite \303\251 indefinido (ou infinito), digite a sua resposta como dne.
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Dois Limites Trigonom\303\251tricos Especiais</Font></Text-field>Os 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 and 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ir\303\243o aparecer repetidamente neste Guia de Estudo. Estes limites ser\303\243o discutidos graficamente neste cap\303\255tulo e rigorosamente derivados com a assist\303\252ncia de alguma geometria no Cap\303\255tulo 4, o Cap\303\255tulo Limites de Fun\303\247\303\265es Trigonom\303\251tricas ( Limits of Trigonometric Functions chapter ). Justifica\303\247\303\243o Gr\303\241fica para estes limites \303\251 obtido das Figuras 1.6 e 1.7. O primeiro limite ilustrado por
plot( sin(theta)/theta, theta=-4*Pi..4*Pi);Figura 1.6 Gr\303\241fico de 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 no intervalo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY3LUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNjctRiM2Oy1JI21uR0YkNjVRKyZ1bWludXMwOzRGJy8lJ2ZhbWlseUdRJnNlcmlmRicvJSVzaXplR1EjMTJGJy8lJWJvbGRHUSZmYWxzZUYnLyUnaXRhbGljR0ZALyUqdW5kZXJsaW5lR0ZALyUqc3Vic2NyaXB0R0ZALyUsc3VwZXJzY3JpcHRHRkAvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lJ29wYXF1ZUdGQC8lK2V4ZWN1dGFibGVHRkAvJSlyZWFkb25seUdGQC8lKWNvbXBvc2VkR0ZALyUqY29udmVydGVkR0ZALyUraW1zZWxlY3RlZEdGQC8lLHBsYWNlaG9sZGVyR0ZALyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR0ZALyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GLDY1USUmcGk7RidGOEY7Rj4vRkJRJXRydWVGJ0ZDRkVGR0ZJRkxGT0ZRRlNGVUZXRllGZW5GZ24vRmpuUSdpdGFsaWNGJy1JI21vR0YkNj5RIixGJ0Y4RjtGPkZBRkNGRUZHRklGTEZPRlFGU0ZVRldGWUZlbkZnbkZpbi8lJmZlbmNlR0ZALyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmdwLUZkbzYtUTEmSW52aXNpYmxlVGltZXM7RidGaW5GZ29GaW9GW3BGXXBGX3BGYXBGY3BGZXBGaHBGanAtRjU2NVEiNEYnRjhGO0Y+RkFGQ0ZFRkdGSUZMRk9GUUZTRlVGV0ZZRmVuRmduRmluRlxvRjhGO0Y+RkFGQ0ZFRkdGSUZMRk9GUUZTRlVGV0ZZRmVuRmduRmluRjhGO0Y+RkFGQ0ZFRkdGSUZMRk9GUUZTRlVGV0ZZRmVuRmduRmluLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnLUZkbzY+USIuRidGOEY7Rj5GQUZDRkVGR0ZJRkxGT0ZRRlNGVUZXRllGZW5GZ25GaW5GZ29GaW9GW3BGXXBGX3BGYXBGY3BGZXBGaHBGOEY7Rj5GQUZDRkVGR0ZJRkxGT0ZRRlNGVUZXRllGZW5GZ25GaW4=
e o segundo limiter por
plot( (1-cos(theta))/theta, theta=-4*Pi..4*Pi);Figure 1.7 Gr\303\241fico de 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 no intervalo 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
Estes limites s\303\243o muito especiais. Qualquer desvio destas formas deve ser tratado com cuidado. Se um limite n\303\243o enquadra exatamente uma destas formas, substitui\303\247\303\265es alg\303\251bricas e simplifica\303\247\303\265es podem ser usadas na tentativa de transformar o limite em um destes limites especiais. O exemplo 1.5 ilustra como um problema pode ser decomposto (broken down) em um destes limites especiais.
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Exemplo 1.5: <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KkYrLUknbXVuZGVyR0YkNiUtSSNtb0dGJDYtUSRsaW1GJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR1EldHJ1ZUYnLyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTY2NjY2N2VtRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHRkgvRjlRJ2l0YWxpY0YnLUY1Ni1RJyZyYXJyO0YnRjhGO0Y+RkBGQkZEL0ZHRj1GSS9GTFEsMC4yNzc3Nzc4ZW1GJy9GT0Zpbi1JI21uR0YkNiRRIjBGJ0Y4RjgvJSxhY2NlbnR1bmRlckdGPUYrLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjRlbUYnLyUmZGVwdGhHRmZvLyUqbGluZWJyZWFrR1ElYXV0b0YnRistRiM2Ji1GNTYtUSIoRidGOC9GPEZIRj4vRkFGSEZCRkRGZ25GSUZLL0ZPRk0tSSZtZnJhY0dGJDYoLUYjNiZGKy1GIzYmLUYsNiVRJHNpbkYnL0ZXRj1GOC1GIzYmRmFwLUYjNiZGKy1GIzYmLUZcbzYkUSIzRidGOC1GNTYtUTEmSW52aXNpYmxlVGltZXM7RidGOEY7Rj5GQEZCRkRGZ25GSUZLRmZwRlNGOEYrRjgtRjU2LVEiKUYnRjhGZHBGPkZlcEZCRkRGZ25GSUZLL0ZPUSwwLjExMTExMTFlbUYnRjhGK0Y4RitGOC1GIzYmRistRiM2Ji1GXG82JFEiNEYnRjhGW3JGU0Y4RitGOC8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGX3MvJSliZXZlbGxlZEdGPUZeckY4RitGOEYrRjg=">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</Equation> </Text-field>A fun\303\247\303\243of := x -> sin(3*x)/(4*x):
'f'(x) = f(x);
n\303\243o \303\251 definida para x = 0, e n\303\272mero nenhum de fatora\303\247\303\243o ou outra manipula\303\247\303\243o pode eliminar a "divis\303\243o por zero" , problema em x = 0. Em situa\303\247\303\265es como esta o gr\303\241fico \303\251 ainda mais importante. A Figura 1.8 mostra um gr\303\241fico de f(x) e tamb\303\251m mostra um "orif\303\255cio" no ponto onde x = 0.JSFH
Figure 1.8Gr\303\241fico de 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mostrando o "orif\303\255cio" no ponto onde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkktRjY2LVEiPUYnRjJGOUY7Rj1GP0ZBRkNGRS9GSFEsMC4yNzc3Nzc4ZW1GJy9GS0ZQRjUtSSNtbkdGJDYkUSIwRidGMkYy.
Figura 1.8 sugere que o limite deveria existir e seus valores deveriam ser em torno de 0.75. Para reunir mais evid\303\252ncias em suportes a estas observa\303\247\303\265es, crie uma tabela ( table ) de valores para a fun\303\247\303\243o avaliada pr\303\263ximo de zero. Valores de f(x) para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI8RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1JI21uR0YkNiRRIjBGJ0Y5RjVGOQ==(\303\240 esquerda do ponto limite) aparecem na Tabela 1.1.
Tabela 1.1 sugere que o limite one-sided pela esquerda existe e tem 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 .Similarlmente, valores de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RidGNEY3Rj5GPkYrRj5GK0Y+ para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI+RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1JI21uR0YkNiRRIjBGJ0Y5Rjk= (\303\240 direita do ponto limite) aparecem na Tabela 1.2.
xR := [seq( a+h, h=H )]:
bodyR := seq( < X | f(X) >, X=xR ):
< header, bodyR >;Tabela 1.2 Valores de 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 para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY5LUkjbWlHRiQ2NVEieEYnLyUnZmFtaWx5R1Emc2VyaWZGJy8lJXNpemVHUSMxMkYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSp1bmRlcmxpbmVHRjcvJSpzdWJzY3JpcHRHRjcvJSxzdXBlcnNjcmlwdEdGNy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnLyUnb3BhcXVlR0Y3LyUrZXhlY3V0YWJsZUdGNy8lKXJlYWRvbmx5R0Y3LyUpY29tcG9zZWRHRjcvJSpjb252ZXJ0ZWRHRjcvJStpbXNlbGVjdGVkR0Y3LyUscGxhY2Vob2xkZXJHRjcvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHRjcvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnL0ZYUSdub3JtYWxGJy8lJmZlbmNlR0Y3LyUqc2VwYXJhdG9yR0Y3LyUpc3RyZXRjaHlHRjcvJSpzeW1tZXRyaWNHRjcvJShsYXJnZW9wR0Y3LyUubW92YWJsZWxpbWl0c0dGNy8lJ2FjY2VudEdGNy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmpvLUZlbjY+USI+RidGL0YyRjUvRjlGN0Y7Rj1GP0ZBRkRGR0ZJRktGTUZPRlFGU0ZVRmhuRmpuRlxvRl5vRmBvRmJvRmRvRmZvRmhvRltwRlotSSNtbkdGJDY1USMwLkYnRi9GMkY1RmBwRjtGPUY/RkFGREZHRklGS0ZNRk9GUUZTRlVGaG5GL0YyRjVGYHBGO0Y9Rj9GQUZERkdGSUZLRk1GT0ZRRlNGVUZobg==
Tabela 1.2 d\303\241 suporte a afirma\303\247\303\243o de que o limite lateral pela direita existe e tem 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 .A exist\303\252ncia e a concord\303\242ncia de ambos os limites laterais, pela esquerda e pela direita ( one-sided ) \303\251 um suporte adicional para a afirma\303\247\303\243o de que o valor do limite lateral \303\251 3/4.
<Text-field style="Heading 3" layout="Heading 3">Nota</Text-field>Este argumento n\303\243o \303\251 uma justifica\303\247\303\243o completa da avalia\303\247\303\243o deste limite.O problema \303\251 que esta tabela de valores da fun\303\247\303\243o nunca \303\251 suficiente para justificar a avalia\303\247\303\243o de um limite. A completa avalia\303\247\303\243o deste limite ser\303\241 discutida no Cap\303\255tilo 4, Cap\303\255tulo Limites de Fun\303\247\303\265es Trigonom\303\251tricas ( Limits of Trigonometric Functions chapter ). Alguns dos problemas que podem surgir do uso de uma tabela ou gr\303\241fico para avaliar um limite s\303\243o discutidos no pr\303\263ximo cap\303\255tulo .Quest\303\243o 1.5Encontre o limite: 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
<Text-field style="Heading 1" layout="Heading 2" spaceabove="7"><Font encoding="UTF-8">Gr\303\241ficos e Tabelas Podem Induzir a Erro</Font></Text-field>Um gr\303\241fico n\303\243o \303\251 sempre suficiente para determinar um limite. Quase todos os gr\303\241ficos gerados por computador s\303\243o, na realidade, desenhos de uma cole\303\247\303\243o de pontos conectados para formar uma curva. Se a cole\303\247\303\243o de pontos n\303\243o for feita prudentemente, algumas informa\303\247\303\265es do desenho sobre a fun\303\247\303\243o ficar\303\243o faltando. Pela mesma argumenta\303\247\303\243o, uma tabela de valores pode levar a erro.Estes obst\303\241culos podem com frequ\303\252ncia ser superados pela sele\303\247\303\243o de uma sess\303\243o de uma janela apropriada da fun\303\247\303\243o ou a sele\303\247\303\243o de um conjunto (set ) de pontos a usar para criar a tabela.Isto \303\251 ilustrado no Exemplo 1.6.
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Exemplo 1.6: <Equation executable="false" style="2D Math" input-equation="" display="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">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KkYrLUknbXVuZGVyR0YkNiUtSSNtb0dGJDYtUSRsaW1GJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR1EldHJ1ZUYnLyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTY2NjY2N2VtRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHRkgvRjlRJ2l0YWxpY0YnLUY1Ni1RJyZyYXJyO0YnRjhGO0Y+RkBGQkZEL0ZHRj1GSS9GTFEsMC4yNzc3Nzc4ZW1GJy9GT0Zpbi1JI21uR0YkNiRRIjBGJ0Y4RjgvJSxhY2NlbnR1bmRlckdGPUYrLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjRlbUYnLyUmZGVwdGhHRmZvLyUqbGluZWJyZWFrR1ElYXV0b0YnRistRiM2Ji1GNTYtUSIoRidGOC9GPEZIRj4vRkFGSEZCRkRGZ25GSUZLL0ZPRk0tSSZtZnJhY0dGJDYoLUYjNiZGKy1JJW1zdXBHRiQ2JS1GIzYmRmFwLUYjNihGKy1GIzYmLUZcbzYkUSQxMDBGJ0Y4LUY1Ni1RMSZJbnZpc2libGVUaW1lcztGJ0Y4RjtGPkZARkJGREZnbkZJRktGZnBGU0Y4LUY1Ni1RKCZtaW51cztGJ0Y4RjtGPkZARkJGREZnbkZJRmhuRmpuLUYjNiYtRiw2JVEkY29zRicvRldGPUY4LUYjNiZGYXAtRiM2JEZTRjgtRjU2LVEiKUYnRjhGZHBGPkZlcEZCRkRGZ25GSUZLL0ZPUSwwLjExMTExMTFlbUYnRjhGK0Y4RitGOEZockY4LUZcbzYkUSIyRidGOC8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGK0Y4LUYjNiQtRlxvNiRRJzEwMDAwMEYnRjhGOC8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGXXQvJSliZXZlbGxlZEdGPUZockY4RitGOEYrRjg=</Equation> </Text-field>Figura 1.9 cont\303\251m um gr\303\241fico da fun\303\247\303\243of := x -> (100*x - cos(x))^2/100000:
'f'(x) = f(x);
junto com o gr\303\241fico de 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 .
Figura 1.9 Exemplo 1.6: As fun\303\247\303\265es 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 (em negro) e 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 (em vermelho).
O gr\303\241fico de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RidGNEY3Rj5GPkYrRj5GK0Y+ \303\251 indistingu\303\255vel do gr\303\241fico de 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 neste intervalo.Tabela 1.3 cont\303\251m valores de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RidGNEY3Rj5GPkYrRj5GK0Y+ computados para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\240 esquerda e \303\240 direita de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=.
O lado esquerdo da Tabela 1.3 mostra valores convergindo para x = 0 pela direita do ponto limite, enquanto o lado direito mostra valores convergindo para x = 0 pela direita do ponto de limiteA mais prov\303\241vel conjectura sobre este limite baseada na Tabela 1.3 \303\251 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Entretanto, isto est\303\241 incorreto.Este limite pode ser avaliado pela avalia\303\247\303\243o direta desta fun\303\247\303\243o em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4= . Esta opera\303\247\303\243o forneceq1 := Limit( f(x), x=0 ) = f(0):
q1;
`` = evalf( f(0) );
o qual est\303\241 em concord\303\242ncia com Maple, como visto em Limit( f(x), x=0 ) = limit(f(x),x=0);
Portanto, Figura 1.9 e Tabela 1.3 induziram a erro. Figura 1.10, obtida ao enfocar no intervalo ( interval ) [-0.01, 0.01 ], mostra que as fun\303\247\303\265es s\303\243o bem diferentes.
Figura 1.10 As fun\303\247\303\265es 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 (em negro) e 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 (em vermelho), graphed no intervalo 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
O objetivo do Exemplo 1.6 \303\251 ilustrar que nem um gr\303\241fico nem uma tabela de valores podem justificar completamente a avalia\303\247\303\243o de um limite.Gr\303\241ficos da fun\303\247\303\243o e tabelas de valores correspondentes podem ser \303\272teis, mas s\303\243o somente o primeiro passo. Felizmente, a informa\303\247\303\243o adicional da tabela de valores da fun\303\247\303\243o sugere outros m\303\251todos e t\303\251cnicas para justificar a avalia\303\247\303\243o de um limite. A rigorosa avalia\303\247\303\243o de limites \303\251 discutida no Cap\303\255tulo 2, o cap\303\255tilo intitulado Defini\303\247\303\243o Precisa de Limite ( Precise Definition of the Limit ).Quest\303\243o 1.5Encontre o limite: 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
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Limites de Fun\303\247\303\265es Oscilat\303\263rias</Font></Text-field>As fun\303\247\303\265es trigonom\303\251tricas fundamentais s\303\243o as fun\303\247\303\265es seno ( sine )e cosseno ( cosine ). Estas fun\303\247\303\265es s\303\243o peri\303\263dicas ( periodic ) (com per\303\255odo ( period )LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLUYsNiVRJyYjOTYwO0YnLyUnaXRhbGljR0Y+RjVGNUYrRjU= ). Isto significa que os valores da fun\303\247\303\243o se repetem a cada LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLUYsNiVRJyYjOTYwO0YnLyUnaXRhbGljR0Y+RjVGNUYrRjU= unidades, como mostra a Figura 1.11.
plot( [sin(x), cos(x)], x=-3*Pi..3*Pi, color=[red,blue],
legend=["y = sin(x)","y = cos(x)"] );Figura 1.11 Tr\303\252s per\303\255odos da fun\303\247\303\265es seno e cosseno.
Todas as fun\303\247\303\265es peri\303\263dicas ( periodic )oscilam. Quando compostas com outras fun\303\247\303\265es, estas oscila\303\247\303\265es podem ser problem\303\241ticas em problemas de limite. Se uma fun\303\247\303\243o oscila com uma amplitude fixa, esta fun\303\247\303\243o n\303\243o converge ( converge ), pois a vari\303\241vel ( variable ) cresce (ou decresce) sem limites (restri\303\247\303\265es). Se, estretanto, a amplitude ( amplitude )decresce, ent\303\243o existe uma chance de o limite existir. (Este tipo de limite ser\303\241 discutido em detalhe no Capitulo 5, o cap\303\255tulo sobre Limites no Infinito ( Limits at Infinity .))Exemplo 1.7 ilustra o limite de uma fun\303\247\303\243o com oscila\303\247\303\265es pr\303\263ximo de um ponto limite finito finite .
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Exemplo 1.7: <Equation executable="false" style="2D Math" input-equation="" display="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">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KkYrLUknbXVuZGVyR0YkNiUtSSNtb0dGJDYtUSRsaW1GJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR1EldHJ1ZUYnLyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTY2NjY2N2VtRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHRkgvRjlRJ2l0YWxpY0YnLUY1Ni1RJyZyYXJyO0YnRjhGO0Y+RkBGQkZEL0ZHRj1GSS9GTFEsMC4yNzc3Nzc4ZW1GJy9GT0Zpbi1JI21uR0YkNiRRIjBGJ0Y4RjgvJSxhY2NlbnR1bmRlckdGPUYrLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjRlbUYnLyUmZGVwdGhHRmZvLyUqbGluZWJyZWFrR1ElYXV0b0YnRistRiM2Ji1GLDYlUSRzaW5GJy9GV0Y9RjgtRiM2Ji1GNTYtUSIoRidGOC9GPEZIRj4vRkFGSEZCRkRGZ25GSUZLL0ZPRk0tRiM2JkYrLUkmbWZyYWNHRiQ2KC1GIzYkLUZcbzYkUSIxRidGOEY4LUYjNiRGU0Y4LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zeci8lKWJldmVsbGVkR0Y9RitGOC1GNTYtUSIpRidGOEZqcEY+RltxRkJGREZnbkZJRksvRk9RLDAuMTExMTExMWVtRidGOEYrRjhGK0Y4RitGOA==</Equation> </Text-field>TA fun\303\247\303\243of := x -> sin(1/x):
'f'(x) = f(x);
n\303\243o \303\251 definida em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4=. Figura 1.12 evita as dificuldades de desenhar o gr\303\241fico pr\303\263ximo \303\240 origem, e mostra que 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 oscila entre LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JS1JI21vR0YkNi1RKCZtaW51cztGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGOi8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjFGJ0Y1RjVGK0Y1 and 1.
plot([[x,f(x),x=-2/Pi..-0.01],[x,f(x),x= 0.01.. 2/Pi]], color=red);Figura 1.12 Gr\303\241fico de 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 no intervalo 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
Observe que o per\303\255odo ( period ) das oscila\303\247\303\265es aumenta quando os pontos se movem pr\303\263ximo \303\240 origem ( origin )-- pela esquerda e pela direita. Isto sugere que a fun\303\247\303\243o tomar\303\241 todos os valores entre LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JS1JI21vR0YkNi1RKCZtaW51cztGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGOi8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjFGJ0Y1RjVGK0Y1 e 1 infinitamente freq\303\274ente pr\303\263ximo a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPi1GOzYtUSIuRidGPkZARkNGRUZHRklGS0ZNL0ZQUSYwLjBlbUYnL0ZTRmZuRj4=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.Observe que a avalia\303\247\303\243o do Maple do limite informa que o limite n\303\243o existe e a raz\303\243o disto \303\251 que os valores oscilam entre LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JS1JI21vR0YkNi1RKCZtaW51cztGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGOi8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjFGJ0Y1RjVGK0Y1 and 1.Limit( f(x), x=0 ) = limit(f(x), x=0);
Quest\303\243o 1.6Encontre o limite: 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Se o limite \303\251 indefinido (ou infinito), digite a sua resposta como dne.
<Text-field style="Heading 1" layout="Heading 2" spaceabove="7">Resumo</Text-field>Neste cap\303\255tulo, desenvolvemos o seguinte procedimento geral para avalia\303\247\303\243o de limites:
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">A Concept-Based Approach to Evaluating <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field>
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7"></Text-field>O primeiro passo na maioria dos problemas de limite seria fazer o gr\303\241fico ou visualizar de outra forma a fun\303\247\303\243o em um intervalo apropriado.Os limites mais f\303\241ceis de computar s\303\243o aqueles onde \303\251 poss\303\255vel avaliar a fun\303\247\303\243o no ponto do limite. O gr\303\241fico pode sugerir se isto \303\251 permiss\303\255vel para um problema.Divis\303\243o por zero \303\251 um problema que pode impedir uma avalia\303\247\303\243o direta de um limite.Fatora\303\247\303\243o e cancelamento de pot\303\252ncias powers of (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEoJm1pbnVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUYsNiVRImFGJ0Y0RjdGPkYrRj4= ) pode levar a um limite equivalente que pode ser computado por avalia\303\247\303\243o direta. (Porque, este m\303\251todo de trabalho muda o valor da fun\303\247\303\243o somente em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRImFGJ0Y0RjdGPkYrRj4= e o valor da fun\303\247\303\243o neste ponto n\303\243o influencia a exist\303\252ncia do limite ou do seu valor).Um limite lateral existe somente quando ambos os limites laterais , pela esquerda e pela direita existam e tenham o mesmo valor. Se ambos os limites laterais pela esquerda e pela direita existirem mas tiverem valores diferentes, a fun\303\247\303\243o experimenta um salto( jump ) neste ponto e o limite lateral n\303\243o existe.Existem dois limites especiais envolvendo fun\303\247\303\265es trigonom\303\251tricas. Estes limites podem ser usados para avaliar outros limites trigonom\303\251tricos, incluindo alguns que aparecer\303\243o na discuss\303\243o de derivadas( derivatives ) na Unidade 2 .N\303\243o seja muito confiante quando usar um gr\303\241fico ou tabela de valores da fun\303\247\303\243o para avaliar um limite. Se n\303\243o forem escolhidos com muito cuidado, eles poder\303\243o induzir a erro!
<Text-field style="Heading 1" layout="Heading 2" spaceabove="7"><Font encoding="UTF-8">Qual \303\251 o Pr\303\263ximo?</Font></Text-field>O pr\303\263ximo cap\303\255tulo usa as id\303\251ias apresentadas neste cap\303\255tulo para desenvolver uma defini\303\247\303\243o precisade um limite.Chapter Navigation1 2345678910111213141516171819202122232425262728293031Next >>
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