<Text-field style="Heading 1" layout="Heading 1">Introdu\303\247\303\243o</Text-field>

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Cap\303\255tulo 3

Regras de Limite

Se todos os limites tivessem de ser avaliados usando a defini\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUScmIzk0OTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEoJm1pbnVzO0YnRjcvJSZmZW5jZUdGNi8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y2LyUqc3ltbWV0cmljR0Y2LyUobGFyZ2VvcEdGNi8lLm1vdmFibGVsaW1pdHNHRjYvJSdhY2NlbnRHRjYvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZOLUYsNiVRJyYjOTQ4O0YnRjRGN0Y3RitGNw==, os limites seriam c\303\241lculos extremamente tediosos. Felizmente, existem algumas regras gerais, chamadas as Regras de Limite, para avalia\303\247\303\243o de limites. Estas leis nos permitem evitar a aplica\303\247\303\243o da defini\303\247\303\243o para cada limite encontrado.

A qualidade mais importante a se desenvolver ao longo deste cap\303\255tulo \303\251 a habilidade para avaliar um limite aplicando as Regras de Limite numa ordem apropriada.

O maplet Limit fornece um excelente ferramenta para aprender os nomes das leis de limites e como elas podem ser aplicadas para avaliar um limite. Isto ser\303\241 demonstrado com tr\303\252s exemplos.

O Teorema do Confronto \303\251 uma lei de limite que \303\251 bastante diferente de todas as outras. Ela depende das informa\303\247\303\265es que se tenha de duas fun\303\247\303\265es relacionadas. Por causa do tipo diferente deste resultado, ela \303\251 apresentada em separado das outras leis. Um exemplo de sua aplica\303\247\303\243o ser\303\241 dado neste cap\303\255tulo; exemplos adicionais ser\303\243o dados no cap\303\255tulo 4.

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">\303\215ndice de Termos</Text-field>

Os seguintes termos, neste cap\303\255tulo, t\303\252m conex\303\243o com o Dicion\303\241rio do Maple; para mais informa\303\247\303\265es v\303\241 em Dicion\303\241rio Matem\303\241tico.

limite superior

limite

constante

denominador

diferen\303\247a

avaliar

fun\303\247\303\243o

gr\303\241fico

hip\303\263tese

inequa\303\247\303\243o

limite

m\303\272ltiplo

unilateral

positivo

expoente

produto

prova

regra do quociente

quociente

seno

regra do sandu\303\255che

tabelas

fun\303\247\303\243o trigonom\303\251trica

bilateral

valor

zero

<Text-field style="Heading 1" layout="Heading 1">Inicializa\303\247\303\243o</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJStleGVjdXRhYmxlR0Y9Rjk=

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">As Regras de Limite</Text-field>

Em geral, os limites podem ser avaliados tomando o limite de termos individuais dentro de uma express\303\243o e depois combinando esses resultados usando as opera\303\247\303\265es matem\303\241ticas apropriadas. Por exemplo, o limite de uma soma \303\251 a soma dos limites. A Tabela 3.1 resume as propriedades fundamentais de limites.

Nome

F\303\263rmula

Condi\303\247\303\265es Extras

Constante

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

Identidade

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

Multiplicada

por Constante

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

Soma

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2KkYrLUknbXVuZGVyR0YkNiUtSSNtb0dGJDYtUSRsaW1GJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR1EldHJ1ZUYnLyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTY2NjY2N2VtRictRiM2Ji1GLDYnUSJ4RicvJSdpdGFsaWNHRkgvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnL0Y5USdpdGFsaWNGJy9JK21zZW1hbnRpY3NHRiRRJGV4cEYnLUY1Ni1RJyZyYXJyO0YnRjhGO0Y+RkBGQkZEL0ZHRj1GSUZLL0ZPRk0tRiw2J1EiYUYnRlZGWEZlbkZnbkY4LyUsYWNjZW50dW5kZXJHRj1GKy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC40ZW1GJy8lJmRlcHRoR0Zpby8lKmxpbmVicmVha0dRJWF1dG9GJ0YrLUYjNiQtSShtZmVuY2VkR0YkNiQtRiM2JkYrLUYjNihGKy1GIzYmLUYsNidRImZGJ0ZWRlhGZW5GZ24tRjU2LVEifkYnRjhGO0Y+RkBGQkZERl1vRklGS0Zeby1GZXA2JC1GIzYkRlNGOEY4RjgtRjU2LVEiK0YnRjhGO0Y+RkBGQkZERl1vRkkvRkxRLDAuMjIyMjIyMmVtRicvRk9GW3ItRiM2Jy1GLDYnUSJnRidGVkZYRmVuRmduLUY1Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRjhGO0Y+RkBGQkZERl1vRklGS0Zeby1GIzYkLUZlcDYkLUYjNiZGK0ZlcUYrRjhGOEY4RitGOEYrRjhGK0Y4RjhGOEYrRjgtRjU2LVEiPUYnRjhGO0Y+RkBGQkZERl1vRkkvRkxRLDAuMjc3Nzc3OGVtRicvRk9GX3MtRiM2KEYrLUYjNilGMUYrRmRvRl1xLUY1Ni5GYnFGWEY4RjtGPkZARkJGREZdb0ZJRktGXm9GZ3JGOEZncS1GIzYqRitGMUYrRmRvRitGXXJGK0Y4RitGOEYrRjg=

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 e 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 existem

Diferen\303\247a

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

Produto

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

Quociente

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Pot\303\252ncia

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Raiz

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 existe, e se n for par, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY3LUkjbWlHRiQ2I1EhRictRiM2PkYrLUknbXVuZGVyR0YkNiUtSSNtb0dGJDY+USRsaW1GJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUlc2l6ZUdRIzEyRicvJSVib2xkR1EmZmFsc2VGJy8lJ2l0YWxpY0dGQC8lKnVuZGVybGluZUdGQC8lKnN1YnNjcmlwdEdGQC8lLHN1cGVyc2NyaXB0R0ZALyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvJSdvcGFxdWVHRkAvJStleGVjdXRhYmxlR0ZALyUpcmVhZG9ubHlHRkAvJSljb21wb3NlZEdGQC8lKmNvbnZlcnRlZEdGQC8lK2ltc2VsZWN0ZWRHRkAvJSxwbGFjZWhvbGRlckdGQC8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdGQC8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdGQC8lKnNlcGFyYXRvckdGQC8lKXN0cmV0Y2h5R0ZALyUqc3ltbWV0cmljR0ZALyUobGFyZ2VvcEdGQC8lLm1vdmFibGVsaW1pdHNHUSV0cnVlRicvJSdhY2NlbnRHRkAvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4xNjY2NjY3ZW1GJy1GIzY5RistSShtYWN0aW9uR0YkNiQtRiw2NVEieEYnRjhGO0Y+L0ZCRmhvRkNGRUZHRklGTEZPRlFGU0ZVRldGWUZlbkZnbi9Gam5RJ2l0YWxpY0YnLyUrYWN0aW9udHlwZUdRNm1hcGxlc29mdDpwbGFjZWhvbGRlckYnLUY1Nj5RJyZyYXJyO0YnRjhGO0Y+RkFGQ0ZFRkdGSUZMRk9GUUZTRlVGV0ZZRmVuRmduRmluRlxvRl5vL0Zhb0Zob0Zib0Zkby9GZ29GQEZpby9GXHBRLDAuMjc3Nzc3OGVtRicvRl9wRmVxLUZkcDYkLUYsNjVRImFGJ0Y4RjtGPkZpcEZDRkVGR0ZJRkxGT0ZRRlNGVUZXRllGZW5GZ25GanBGXHFGK0Y4RjtGPkZBRkNGRUZHRklGTEZPRlFGU0ZVRldGWUZlbkZnbkZpbi8lLGFjY2VudHVuZGVyR0ZALUY1Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRmluRlxvRl5vRmBvRmJvRmRvRmNxRmlvRltwL0ZfcEZdcC1GZHA2JC1GLDY1USJmRidGOEY7Rj5GaXBGQ0ZFRkdGSUZMRk9GUUZTRlVGV0ZZRmVuRmduRmpwRlxxRistSShtZmVuY2VkR0YkNjUtRiM2NS1GLDY2RmhwRjhGO0Y+RmlwRkNGRUZHRklGTEZPRlFGU0ZVRldGWUZlbkZnbkZqcC9JK21zZW1hbnRpY3NHRiRRJGV4cEYnRjhGO0Y+RkFGQ0ZFRkdGSUZMRk9GUUZTRlVGV0ZZRmVuRmduRmluRjhGO0Y+RkFGQ0ZFRkdGSUZMRk9GUUZTRlVGV0ZZRmVuRmduRmluLUY1Ni1RMSZJbnZpc2libGVUaW1lcztGJ0ZpbkZcb0Zeb0Zgb0Zib0Zkb0ZjcUZpb0ZbcEZhci1GNTY+USI+RidGOEY7Rj5GQUZDRkVGR0ZJRkxGT0ZRRlNGVUZXRllGZW5GZ25GaW5GXG9GXm9GYG9GYm9GZG9GY3FGaW9GW3BGYXJGYXMtSSNtbkdGJDY1USIwRidGOEY7Rj5GQUZDRkVGR0ZJRkxGT0ZRRlNGVUZXRllGZW5GZ25GaW5GOEY7Rj5GQUZDRkVGR0ZJRkxGT0ZRRlNGVUZXRllGZW5GZ25GaW5GK0Y4RjtGPkZBRkNGRUZHRklGTEZPRlFGU0ZVRldGWUZlbkZnbkZpbg==

Tabela 3.1 Propriedades fundamentais de limites

Note as condi\303\247\303\265es listadas na terceira coluna. Se as condi\303\247\303\265es para uma regra n\303\243o forem satisfeitas, a regra n\303\243o pode ser usada para justificar a avalia\303\247\303\243o de um limite.

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Exemplos</Text-field>

O tr\303\252s exemplos seguintes demonstram o uso destas regras de limite na avalia\303\247\303\243o de limites. Exemplos 3.1 e 3.2 foram tiradas do Cap\303\255tulo 2; O Exemplo 3.3 \303\251 novo.

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Exemplo 3.1: Use as Regras de Limite para avaliar <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field>

Manipula\303\247\303\243o Alg\303\251brica

Observa\303\247\303\265es

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

limite original

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

regra da diferen\303\247a aplicada

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

regra da constante aplicada

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

regra constante multiplicativa aplicada

regra da identidade

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEiPUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZDLUkjbW5HRiQ2JVEiMkYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0YvRi8=

aritm\303\251tica b\303\241sica \342\207\222 resposta final

Tabela 3.2

Passos levando ao c\303\241lculo deLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2K0YrLUknbXVuZGVyR0YkNiUtSSNtb0dGJDYtUSRsaW1GJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR1EldHJ1ZUYnLyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTY2NjY2N2VtRictRiM2KEYrLUkobWFjdGlvbkdGJDYkLUYsNiVRInhGJy8lJ2l0YWxpY0dGSC9GOVEnaXRhbGljRicvJSthY3Rpb250eXBlR1E2bWFwbGVzb2Z0OnBsYWNlaG9sZGVyRictRjU2LVEnJnJhcnI7RidGOEY7Rj4vRkFGSEZCRkQvRkdGPUZJL0ZMUSwwLjI3Nzc3NzhlbUYnL0ZPRmBvLUkjbW5HRiQ2JFEiM0YnRjhGK0Y4LyUsYWNjZW50dW5kZXJHRj1GKy1JKG1mZW5jZWRHRiQ2JC1GIzYqLUZjbzYkUSIyRidGOC1GNTYtUTEmSW52aXNpYmxlVGltZXM7RidGOEY7Rj5GQEZCRkRGXm9GSUZLL0ZPRk0tRiw2JkZYRllGZW4vSSttc2VtYW50aWNzR0YkUSRleHBGJ0ZgcC1GNTYtUSgmbWludXM7RidGOEY7Rj5GQEZCRkRGXm9GSUZLRmNwRmBwLUZjbzYkUSI0RidGOEY4RjhGYHAtRjU2LVEiPUYnRjhGO0Y+RkBGQkZERl5vRklGS0ZjcEZgcEZdcEY4RitGOA==

A tabela 3.2 mostra a sequ\303\252ncia de passos levando \303\240 conclus\303\243o 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. Como cada uma das Regras de Limite na Tabela 3.1 \303\251 na verdade um teorema sobre limites, uma aplica\303\247\303\243o correta destas leis constitui uma prova que 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.

Em adi\303\247\303\243o \303\240 argumenta\303\247\303\243o embutida na lista de passos apresentada, voc\303\252 est\303\241 em condi\303\247\303\265es de repetir estes exemplos no maplet Limit. Este maplet \303\251 na verdade o j\303\241 imbutido Tutor de Limite Passo-a-Passo, acess\303\255vel a partir do menu Ferramentas/Tutores/C\303\241lculo-\303\232nica Vari\303\241vel como M\303\251todos de Limite...

Para especificar um problema no maplet Limit, note que a linha superior deste maplet cont\303\251m campos para a fun\303\247\303\243o, vari\303\241vel, ponto de limite, e se o limite \303\251 bilateral (em branco) ou unilateral \303\240 esquerda ou \303\240 direita. Pressione o bot\303\243o Come\303\247ar, depois aplique as regras de limite clicando o bot\303\243o correspondente no maplet Limit . A barra de menus fornece um resumo de cada regra conhecida (Defini\303\247\303\243o da Regra), ajuda, e um outro modo de aplicar as regras (Aplicar a Regra). Note que a regra selecionada \303\251 geralmente aplicada para a primeira ocorr\303\252ncia; pode ser necess\303\241rio aplicar uma regra muitas vezes seguidas. Figura 3.1 mostra este Tutor inicializado para o exemplo 3.1

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Figure 3.1 O maplet Limit inicializado para Exemplo 3.1

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Exemplo 3.2: Use as Regras de Limite para avaliar<Equation executable="false" style="2D Math_4" input-equation="" display="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">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</Equation> </Text-field>

Manipula\303\247\303\243o Alg\303\251brica

Coment\303\241rios

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

limite original

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

regras de diferen\303\247a e soma aplicadas

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

regras de pot\303\252ncia e da multiplica\303\247\303\243o por constante aplicadas

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

regras da identidade e da constante aplicadas

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEiPUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZDLUkjbW5HRiQ2JVEiNEYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0YvRi8=

aritm\303\251tica b\303\241sica => resposta final

Tabela 3.3 Principais passos para computa\303\247\303\243o de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUkjbWlHRiQ2I1EhRictSSdtdW5kZXJHRiQ2JS1JI21vR0YkNi1RJGxpbUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOy8lKXN0cmV0Y2h5R0Y7LyUqc3ltbWV0cmljR0Y7LyUobGFyZ2VvcEdGOy8lLm1vdmFibGVsaW1pdHNHUSV0cnVlRicvJSdhY2NlbnRHRjsvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4xNjY2NjY3ZW1GJy1GIzYmLUYsNidRInhGJy8lJ2l0YWxpY0dGRi8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjdRJ2l0YWxpY0YnL0krbXNlbWFudGljc0dGJFEkZXhwRictRjM2LVEnJnJhcnI7RidGNkY5RjxGPkZARkIvRkVGO0ZHRkkvRk1GSy1JI21uR0YkNiVRIjFGJ0ZWRjZGNi8lLGFjY2VudHVuZGVyR0Y7RistSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuNGVtRicvJSZkZXB0aEdGaG8vJSpsaW5lYnJlYWtHUSVhdXRvRidGKy1GIzYmLUYzNi1RIihGJ0Y2L0Y6RkZGPC9GP0ZGRkBGQkZbb0ZHRklGXG8tRiM2KUYrLUYjNiZGKy1JJW1zdXBHRiQ2JUZRLUZebzYlUSIyRidGVkY2LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YrRjYtRjM2LVEoJm1pbnVzO0YnRjZGOUY8Rj5GQEZCRltvRkdGSUZcby1GIzYmLUZebzYlUSIzRidGVkY2LUYzNi1RMSZJbnZpc2libGVUaW1lcztGJ0Y2RjlGPEY+RkBGQkZbb0ZHRklGXG9GUUY2LUYzNi1RIitGJ0Y2RjlGPEY+RkBGQkZbb0ZHL0ZKUSwwLjIyMjIyMjJlbUYnL0ZNRmRyLUZebzYlUSI2RidGVkY2RjYtRjM2LVEiKUYnRjZGZnBGPEZncEZARkJGW29GR0ZJL0ZNUSwwLjExMTExMTFlbUYnRjZGXXItRjM2LVEiPUYnRjZGOUY8Rj5GQEZCRltvRkdGSUZcb0Zdci1GXm82JFEiNEYnRjZGNg==

Tabela 3.3 resume os passos principais para a conclus\303\243o de que 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Como cada uma das Regras de Limite na Tabela 3.1 \303\251 na verdade um teorema sobre limites, a aplica\303\247\303\243o correta destas leis constitui uma prova v\303\241lida de que 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 .

O exemplo final envolve a regra do quociente. Note que o maplet Limit aplica a regra selecionada sem checar nenhuma hip\303\263tese. Neste caso a regra do quociente \303\251 aplicada sem verificar se o denominador \303\251 zero.

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Example 3.3: Use as Regras de Limite para avaliar<Equation executable="false" style="2D Math_6" input-equation="" display="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">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</Equation> </Text-field>

Este problema usar\303\241 as regras do produto e do quociente. Seguindo os passos nos Exemplos 3.1 e 3.2, facilmente verifica-se que:

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNipGKy1JJ211bmRlckdGJDYlLUkjbW9HRiQ2LVEkbGltRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dRJXRydWVGJy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjE2NjY2NjdlbUYnLUYjNictRiw2JVEidEYnLyUnaXRhbGljR0ZKL0Y7USdpdGFsaWNGJy1GNzYtUScmcmFycjtGJ0Y6Rj1GQEZCRkRGRi9GSUY/RktGTS9GUUZPLUYjNiUtRjc2LVEoJm1pbnVzO0YnRjpGPUZARkJGREZGRmluRktGTUZqbi1JI21uR0YkNiRRIjJGJ0Y6RjpGK0Y6LyUsYWNjZW50dW5kZXJHRj9GKy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC40ZW1GJy8lJmRlcHRoR0ZbcC8lKmxpbmVicmVha0dRJWF1dG9GJ0YrLUYjNiYtRjc2LVEiKEYnRjovRj5GSkZAL0ZDRkpGREZGRmluRktGTUZqbi1GIzYpRistRiM2JkYrLUklbXN1cEdGJDYlRlUtRmFvNiRRIjRGJ0Y6LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YrRjotRjc2LVEiK0YnRjpGPUZARkJGREZGRmluRksvRk5RLDAuMjIyMjIyMmVtRicvRlFGXHItRiM2J0ZicS1GNzYtUTEmSW52aXNpYmxlVGltZXM7RidGOkY9RkBGQkZERkZGaW5GS0ZNRmpuLUYjNiZGKy1GYHE2JUZVRmBvRmVxRitGOkYrRjpGaHFGYnFGOi1GNzYtUSIpRidGOkZpcEZARmpwRkRGRkZpbkZLRk0vRlFRLDAuMTExMTExMWVtRidGOkYrRjotRjc2LVEiPUYnRjpGPUZARkJGREZGRmluRksvRk5RLDAuMjc3Nzc3OGVtRicvRlFGYHMtRmFvNiRRIzM2RidGOkY6RitGOg==

Pelo fato dos dois limites iniciais existirem, a Regra do Produto pode ser aplicada para se obter:

Agora, como esses dois limites existem e pelo fato que:

a lei do Quociente pode ser aplicada. O resultado \303\251 que:

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<Text-field style="Heading 1" readonly="true" layout="Heading 1">Maple T.A. Maple-Graded Question</Text-field>

Using the Maple-graded question type, you create a question that uses Maple functionality when creating the correct answer and determining if the student's response is correct. The Maple-graded question type has an option to plot the student's response. To include the plot option, insert the corresponding task template. For information on creating questions, making assignments, and exporting Maple T.A. course modules, see Overview of Maple T.A. Integration.

<Text-field style="Heading 2" layout="Heading 2">Algorithmic Variables</Text-field>

Algorithmic variables can be used to create a question that contains randomly generated parameters. You can refer to algorithmic variables in the Question, Answer, Hints, and Feedback input regions, as well as in other variable definitions. To reference an algorithmic variable, include "`$variable`" as text or 2-D Math, for example, x := `$myvariable`. Adding algorithmic variables to the question is optional. In the following execution group, add an algorithmic variable. To include more variables, add more execution groups. If your assignment contains more than one algorithmic variable, ensure the variable names are unique.

randomize(): c1 := RandomTools[Generate](integer(range=-3..3)): while (c1 = 0) do c1 := RandomTools[Generate](integer(range=-3..3)): end do: a1 := RandomTools[Generate](integer(range=1..8)): b1 := RandomTools[Generate](integer(range=-3..3)): ff1 := `if`(b1<>0,a1*x^b1,a1*x^2+(a1-4)*x+(a1/2)); fn1 := Limit((f(x)-f(c1))/(x-c1),x=c1); ans1 := limit((ff1-eval(ff1,x=c1))/(x-c1),x=c1);

LCQqJEkieEc2IiIiIyIiKQ==

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IiM7

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<Text-field style="Heading 1" layout="Heading 1">Quest\303\243o 3.1</Text-field>

The question can contain text, 2-D Math, plots, images, and algorithmic variables. To insert a plot in the question, use the Plot Component. In the following text region, enter the full contents of the question.

Seja f(x) = `$ff1` Encontre `$fn1`

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<Text-field style="Heading 2" layout="Heading 2">Answer</Text-field>

To evaluate a student's response, enter the correct answer. Complete each line of code with a semicolon. The last line must evaluate to an expression that will be stored as the variable `$ANSWER`. Enter Maple code in 1-D Math that evaluates to the correct answer. Add additional execution groups as needed.

ans1;

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<Text-field style="Heading 2" layout="Heading 2">Evaluate Student's Response</Text-field>

To evaluate a student's response, you must evalute to a Boolean value (true or false). The variable `$RESPONSE` is automatically initialized to the student's response. Using `$ANSWER` (the correct answer) and `$RESPONSE` (the student response), one example of evaluating to a Boolean value is, evalb(`$ANSWER`-`$RESPONSE`=0); Enter Maple code to evaluate a student's response. To add additional Maple functionality, use the Answer input region.

evalb(`$ANSWER`-`$RESPONSE`=0);

<Text-field style="Heading 2" layout="Heading 2">Hints</Text-field>

A question can offer hints to students when requested. Hints can contain text, 2-D Math, and algorithmic variables. Adding hints to the question is optional. Enter your hints in the following table. You can provide up to three hints.

Simplify the expression so that the singularity is removed.

Replace this text with hint 2.

Replace this text with hint 3.

with(StringTools): with(DocumentTools): response := GetProperty(`TextArea0`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>Simplify the expression so that the singularity is removed.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet);

<Text-field style="Heading 2" layout="Heading 2">Feedback</Text-field>

A question can display feedback when a student provides an incorrect response. The feedback can contain text, 2-D Math, and algorithmic variables. The feedback can also refer to `$RESPONSE` (the student response) and `$ANSWER` (the correct answer). Adding feedback to the question is optional. In the following text region, enter feedback to an incorrect response.

Replace this text with feedback.

with(StringTools): with(DocumentTools): feedback:=""; StudentResponse := GetProperty(`TextArea0`, `value`); ANSWER := ans1; if evalb(ANSWER-eval(parse(StudentResponse))=0) then feedback:=cat("<math><mrow><mtext mathvariant='bold'>Correct!&NewLine;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext></mrow></math>"); else if StudentResponse = "" then StudentResponse:="<NO RESPONSE>"; fi; feedback:=cat("<math><mrow><mtext mathvariant='bold'>Your answer is incorrect. </mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, "<mtext mathvariant='bold'>Correct Answer: </mtext><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>16</mn></mrow></mstyle></math><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, SubstituteAll("<mrow><mtext>&NewLine;</mtext></mrow>", "`$RESPONSE`", StudentResponse)); feedback:=SubstituteAll(feedback, "`$ANSWER`", "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>16</mn></mrow></mstyle></math>"); feedback:=cat(feedback, "</mrow></math>"); fi; with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Feedback", [[MathMLViewer('value' = feedback, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): if feedback <> "" then Maplets[Display](maplet); fi;

<Text-field style="Heading 1" readonly="true" layout="Heading 1">Maple T.A. Maple-Graded Question</Text-field>

<Text-field style="Heading 2" layout="Heading 2">Algorithmic Variables</Text-field>

with(RandomTools): randomize(): Lf := Limit(f(x),x=c)=L; Lg := Limit(g(x),x=c)=L; n2 := RandomTools[Generate](integer(range=1..3)): d2 := RandomTools[Generate](integer(range=-1..1)): q2 := (f(x)^2+n2*g(x)^2)/(f(x)+d2*g(x)); fn2 := Limit(q2,x=c); ans21 := `if`([n2,d2]=[1,1],L,`if`([n2,d2]=[1,-1],undefined,`if`([n2,d2]=[-1,1],0,`if`([n2,d2]=[-1,-1],2*L,`if`([n2,d2]=[1,0],2*L,`if`([n2,d2]=[-1,0],0,`if`([n2,d2]=[0,0],L,`if`([n2,d2]=[0,1],L/2,undefined)))))))); if (ans21 = undefined) then ans2 := 'dne'; else ans2 := ans21; end if;

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SSp1bmRlZmluZWRHJSpwcm90ZWN0ZWRH

SSRkbmVHNiI=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">Quest\303\243o 3.2</Text-field>

Se `$Lf` e `$Lg`, encontre `$fn2` Se o limite n\303\243o existe, entre sua resposta como dne.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 2" layout="Heading 2">Answer</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=ans2;

<Text-field style="Heading 2" layout="Heading 2">Evaluate Student's Response</Text-field>

if ((`$RESPONSE` = undefined) and evalb(`$RESPONSE` = `$ans21`) = true) then rrr2 := 'dne': else rrr2 := `$RESPONSE`: end if: evalb(`$ANSWER` = rrr2);

<Text-field style="Heading 2" layout="Heading 2">Hints</Text-field>

If the limit of the denominator is not zero, then the problem is trivial.

If the limits of the numerator and denominator are both zero, then you must attempt to simplify the quotient to eliminate this problem.

If the limit of the denominator is zero and the limit of the numerator is not zero, then there is not much you can do. What does this mean about the limit of the quotient?

with(StringTools): with(DocumentTools): response := GetProperty(`TextArea1`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>If the limit of the denominator is not zero, then the problem is trivial.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet);

with(StringTools): with(DocumentTools): response := GetProperty(`TextArea1`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>If the limits of the numerator and denominator are both zero, then you must attempt to simplify the quotient to eliminate this problem.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet);

with(StringTools): with(DocumentTools): response := GetProperty(`TextArea1`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>If the limit of the denominator is zero and the limit of the numerator is not zero, then there is not much you can do. What does this mean about the limit of the quotient?</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet);

<Text-field style="Heading 2" layout="Heading 2">Feedback</Text-field>

Replace this text with feedback.

with(StringTools): with(DocumentTools): feedback:=""; StudentResponse := GetProperty(`TextArea1`, `value`); ANSWER := ans2; if ((eval(parse(StudentResponse)) = undefined) and evalb(eval(parse(StudentResponse)) = ans21) = true) then rrr2 := 'dne':else rrr2 := eval(parse(StudentResponse)):end if: if evalb(ANSWER = rrr2) then feedback:=cat("<math><mrow><mtext mathvariant='bold'>Correct!&NewLine;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext></mrow></math>"); else if StudentResponse = "" then StudentResponse:="<NO RESPONSE>"; fi; feedback:=cat("<math><mrow><mtext mathvariant='bold'>Your answer is incorrect. </mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, "<mtext mathvariant='bold'>Correct Answer: </mtext><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dne</mi></mrow></mstyle></math><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, SubstituteAll("<mrow><mtext>&NewLine;</mtext></mrow>", "`$RESPONSE`", StudentResponse)); feedback:=SubstituteAll(feedback, "`$ANSWER`", "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dne</mi></mrow></mstyle></math>"); feedback:=cat(feedback, "</mrow></math>"); fi; with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Feedback", [[MathMLViewer('value' = feedback, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): if feedback <> "" then Maplets[Display](maplet); fi;

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">O Teorema do Confronto</Text-field>

O limite 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 envolve um produto, mas a Lei do Produto n\303\243o pode ser aplicada porque 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 n\303\243o existe (veja Exemplo 7 no Cap\303\255tulo 1, o cap\303\255tulo Limite de Forma Intuitiva). Isto \303\251 particularmente frustrante porque seria bom se pud\303\251ssemos argumentar que 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 e 0*(nada) = 0. Mas, isto \303\251 verdade somente se soubermos alguma coisa a mais sobre "nada".

O Teorema do Confronto \303\251 uma Lei de Limite que pode ser usada neste tipo de situa\303\247\303\243o.

O Teorema do Confronto

Seja f, g, e h fun\303\247\303\265es satisfazendo

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

para qualquer LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= pr\303\263ximo de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= (exceto possivelmente em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRImFGJ0Y0RjdGPkYrRj4=).

Se 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 e 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 , ent\303\243o 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 .

A ideia por tr\303\241s do Teorema do Confronto \303\251 esta: se voc\303\252 sabe que os valores de uma fun\303\247\303\243o est\303\243o limitados acima e abaixo por fun\303\247\303\265es com limites conhecidos em um ponto --- e estes limites t\303\252m o mesmo valor --- ent\303\243o o limite desconhecido existe e tem o mesmo valor dos outros limites.

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Exemplo 3.4: <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> </Text-field>

Como notou-se no come\303\247o deste cap\303\255tulo, como 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 n\303\243o existe, este limite n\303\243o pode ser calculado usando a Regra do Produto .

Figura 3.2 sugere o que est\303\241 acontecendo para este limite.

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Figura 3.2 Gr\303\241fico de 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 no intervalo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21vR0YkNi1RIltGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJXRydWVGJy8lKnNlcGFyYXRvckdRJmZhbHNlRicvJSlzdHJldGNoeUdGOi8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSi1GIzYoRistRiM2Ji1GMjYtUSgmbWludXM7RidGNS9GOUY9RjsvRj9GPUZARkJGREZGRkhGSy1JJm1mcmFjR0YkNigtRiM2JC1JI21uR0YkNiRRIjFGJ0Y1RjUtRiM2JC1GZm42JFEiMkYnRjVGNS8lLmxpbmV0aGlja25lc3NHRmhuLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmJvLyUpYmV2ZWxsZWRHRj1GK0Y1LUYyNi1RIixGJ0Y1RlQvRjxGOkZVRkBGQkZERkZGSC9GTFEsMC4zMzMzMzMzZW1GJ0ZWRitGNS1GMjYtUSJdRidGNUY4RjtGPkZARkJGREZGRkhGS0Y1RitGNQ==

As oscila\303\247\303\265es na fun\303\247\303\243o seno s\303\243o a raz\303\243o para que 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 n\303\243o exista. As oscila\303\247\303\265es no gr\303\241fico acima significam que 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 n\303\243o exista?

Como todos os valores da fun\303\247\303\243o seno est\303\243o entre -1 e 1 (inclusive), as seguintes inequa\303\247\303\265es devem valer para todos LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RJSZuZTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5LUY2Ni1RIjpGJ0Y5RjtGPkZARkJGREZGRkhGSkZNRjk=

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

(3.1)

A figura 3.3 demonstra estes limites em 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.

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Figura 3.3 Representa\303\247\303\243o gr\303\241fica dos limites em (3.1)

Multiplicando estas inequa\303\247\303\265es por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUklbXN1cEdGJDYlLUYsNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtbkdGJDYkUSIyRicvRjtRJ25vcm1hbEYnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YrRkFGK0ZB (uma quantidade positiva para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVElJm5lO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4= ) resulta em:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUkjbW9HRiQ2LVElJmxlO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZJLUYjNiZGKy1JJW1zdXBHRiQ2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjZRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiMkYnRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRitGNUYrRjVGK0Y1

(3.2)

que vale para todos os LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVElJm5lO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4=.

A figura 3.4 ilustra estes limites em 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.

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Figura 3.4 Gr\303\241fico mostrando que 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 perto de 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

Agora, como 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 e 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 --- note que os dois limites existem e possuem o mesmo limite --- o Teorema do Confrontonos permite a concluir que

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

Podemos confirmar este resultado atrav\303\251s do pr\303\263prio Maple

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

Ly1JJkxpbWl0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJilJInhHRigiIiMiIiItSSRzaW5HRiU2IyokRiwhIiJGLi9GLCIiIUY1

Exemplos adicionais envolvendo o Teorema do Confronto aparecer\303\243o no pr\303\263ximo cap\303\255tulo nos limites trigonom\303\251tricos.

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Resumo</Text-field>

As Regras de Limite, incluindo o Teorema do Confronto, fornecem ferramentas rigorosas para avaliar limites sem ter de fornecer uma prova LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUScmIzk0OTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEoJm1pbnVzO0YnRjcvJSZmZW5jZUdGNi8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y2LyUqc3ltbWV0cmljR0Y2LyUobGFyZ2VvcEdGNi8lLm1vdmFibGVsaW1pdHNHRjYvJSdhY2NlbnRHRjYvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZOLUYsNiVRJyYjOTQ4O0YnRjRGN0Y3RitGNw== para cada e todo limite que encontrarmos. A qualidade mais importante a ser desenvolvida ao longo deste cap\303\255tulo \303\251 a habilidade para aplicar as leis de limite em uma ordem apropriada para se avaliar um limite.

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Pr\303\263ximo Cap\303\255tulo</Text-field>

O pr\303\263ximo cap\303\255tulo ser\303\241 a primeira aplica\303\247\303\243o extensiva das Regras de Limite.

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