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

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Guia de Estudo de C\303\241lculo

\302\251 Maplesoft, a division of Waterloo Maple Inc., 2005

Cap\303\255tulo 2: Defini\303\247\303\243o Precisa de um Limite

O cap\303\255tulo 1 deu uma introdu\303\247\303\243o intuitiva da no\303\247\303\243o de um limite matem\303\241tico. Esta intui\303\247\303\243o foi tirada de

JSFHuma defini\303\247\303\243o informal. De fato, a declara\303\247\303\243o 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(difference) entre

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 e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= pode ser declarada t\303\243o pequena quanto se desejar para todos os valores ( values ) de x suficientemente pr\303\263ximos a-- mas diferentes de -- a.

O desenvolvimento de ferramentas gerais para trabalhar com limites e a aplica\303\247\303\243o deles ( limits ) requer uma defini\303\247\303\243o mais precisa de limite. Este cap\303\255tulo d\303\241 uma defini\303\247\303\243o formal de limite, uma defini\303\247\303\243o que forne\303\247a uma base(estrutura?) na qual somos capazes de desenvolver uma gama imensa de t\303\251cnicas \303\272teis.

O primeiro passo \303\251 desenvolver uma interpreta\303\247\303\243o gr\303\241fica dessa defini\303\247\303\243o precisa. Depois, as manipula\303\247\303\265es alg\303\251bricas envolvidas em aplicar a defini\303\247\303\243o exata ser\303\241 explorada em quatro exemplos.

O cap\303\255tulo conclui com uma defini\303\247\303\243o mais precisa de limites laterais one-sided , vistos no cap\303\255tilo 1.

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Gloss\303\241rio</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).

absolutevalue

assumption

upperbound

constant

constraint

delta

denominator

difference

domain

epsilon

even

factor

function

graph

independent variable

inequality

limit

linear

negative

numerator

one-sided

positive

exponent

product

proof

quadratic

quotient

range

rationalfunction

reciprocal

satisfy

solve

tables

two-sided

value

work

zero

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Inicializa\303\247\303\243o</Text-field>

restart; with( plots ):

<Text-field style="Heading 1" layout="Heading 1">Defini\303\247\303\243o Precisa de Limite:</Text-field>

Come\303\247amos o Cap\303\255tulo 2 com uma defini\303\247\303\243o precisa da fun\303\247\303\243o real-valued LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RidGNEY3Rj5GPkY+RitGPg==. O limite \303\251 um n\303\272mero

designado pela letra L. A nota\303\247\303\243o para o limite de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RidGNEY3Rj5GPkY+RitGPg== em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJhRidGL0YyRjk= \303\251

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

Se este limite \303\251 o n\303\272mero L, escrevemos

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

A defini\303\247\303\243o precisa do limite nos diz quando uma afirma\303\247\303\243o \303\251 verdadeira. Portanto, temos

Defini\303\247\303\243o 2.1

JSFH

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 se e somente se para cada LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy > 0 existir um n\303\272mero LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy > 0 tal que

Esta defini\303\247\303\243o caracteriza rigorosamente a ess\303\252ncia de "proximidade". Ela declara que se os 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

ent\303\243o L \303\251 o limite desses valores da fun\303\247\303\243o. A proximidade dos valores 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. As diferen\303\247as 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\316\265 para todo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2J1EieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lJ29wYXF1ZUdGMS8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjhRJ25vcm1hbEYn pr\303\263ximo de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYvLUkjbWlHRiQ2J1EieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lJ29wYXF1ZUdGMS8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSI9RidGMi9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GLDYnUSJhRidGL0YyRjVGNy1GOzYtUSIuRidGPkZARkNGRUZHRklGS0ZNL0ZQUSYwLjBlbUYnL0ZTRmVuLUY7Ni1RMSZJbnZpc2libGVUaW1lcztGJ0Y+RkAvRkRGMUZFRkdGSUZLRk1GWi9GU1EsMC4zMzMzMzMzZW1GJy1GLDYlUSJBRicvRjBGQkY+RmduLUYsNiVRLHByb3hpbWlkYWRlRidGYG9GPkZnbi1GLDYlUSNkZUYnRmBvRj5GZ24tRiw2JUYuRi9GN0Y+

a a \303\251 ditada pela inequa\303\247\303\243o 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

Como esta defini\303\247\303\243o pode ser dif\303\255cil \303\240 primeira vista (difficult to fathom at first), veremos a seguir v\303\241rios exemplos que interpretam esta defini\303\247\303\243o graficamente.

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Interpreta\303\247\303\243o Gr\303\241fica de Limite</Text-field>

Esta defini\303\247\303\243o exata parece um tanto imponente. A defini\303\247\303\243o torna-se muito mais compreens\303\255vel quando ela pode ser expressa graficamente. O EpsilonDelta maplet prov\303\252 uma excelente interface para a visualiza\303\247\303\243o dos pap\303\251is de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRidGMg== e para obter, como conseq\303\274\303\252ncia, uma compreens\303\243o te\303\263rica de limites.

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Examplo 2.1: <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> </Text-field>

1. Siga os seguinte passos:

Inicie o EpsilonDelta maplet .

Na parte superior da interface, entre em function com 3*x - 4, e entre a = 3, e L = 5 nas janelas apropriadas.

Declare para os limites do gr\303\241fico xmin = 0, xmax = 5, ymin = 0, e ymax = 10.

Declare epsilon = 0.50 e delta = 0.50 (entre com estes valores nas caixas ou use o ponteiro at\303\251 pr\303\263ximo destes valores)

Clique no bot\303\243o Plot na parte inferior da janela Maplet .

Figura 2.1 mostra como fica o maplet ap\303\263s esses passos.

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Figure 2.1 Maplet depois dos passos iniciais detalhados acima .

Note que a faixa horizontal azul n\303\243o est\303\241 contida entre as linhas vermelhas. Isso significa que esse valor de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy \303\251 grande demais para satisfazer (satisfy) as condi\303\247\303\265es da defini\303\247\303\243o de limite.

JSFH

2. Mudar o valor de delta para 0.05.

Agora, a faixa horizontal azul fica inteiramente contida entre as linhas vermelhas. Esse valor de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy satisfaz as condi\303\247\303\265es na defini\303\247\303\243o de limite.

Por\303\251m isso n\303\243o \303\251 suficiente! A defini\303\247\303\243o declara que deve ser poss\303\255vel achar um \316\264 para cada LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy > 0.

At\303\251 o momento, n\303\263s mostramos que essa propriedade \303\251 verdadeira somente para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy = 0.5.

3. Encontre(exatamente ou aproximadamente) o maior valor de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy para o qual as condi\303\247\303\265es da defini\303\247\303\243o do limite sejam satisfeitas para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy = 0.50.

4. Repita o passo 3. para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy = 0.25. (Voc\303\252 deve tentar LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy = 0.1 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy =0.05 para dar in\303\255cio)

5. Voc\303\252 pode fazer algumas observa\303\247\303\265es sobre uma regra geral para a sele\303\247\303\243o de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy para qualquer dado LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy ?

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Exemplo 2.2: <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> </Text-field>

1. Considere o mesmo limite, mas com um valor do limite incorreto (digamos, L = 2).

2. Escolha um valor (positivo)para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy .

3.Voc\303\252 pode encontrar um valor de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy que satisfa\303\247a as condi\303\247\303\265es de um limite?

Se voc\303\252 encontrar um valor de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy > 0 para o qual as condi\303\247\303\265es n\303\243o s\303\243o satisfeitas, isso prova que esse valor de L \303\251 incorreto. Ou o limite tem um valor diferente ou realmente n\303\243o existe.

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Exemplo 2.3: <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> </Text-field>

1. Encontre o valor desse limite.

2. Encontre valores de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy que satisfa\303\247am as condi\303\247\303\265es da defini\303\247\303\243o para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy = 0.50, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy = 0.25, e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy = 0.10.

3. Tente encontrar uma f\303\263mula geral para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy em termos de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy que satisfa\303\247a as condi\303\247\303\265es da defini\303\247\303\243o.

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Exemplo 2.4: <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> </Text-field>

1. Escolha um valor para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy com LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy > 1. Explique porque qualquer LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy > 0 satisfaz as condi\303\247\303\265es na defini\303\247\303\243o.

2. Declare LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy = 0.50. Qual \303\251 o menor valor que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy pode ter para satisfazer as condi\303\247\303\265es na defini\303\247\303\243o?

3. Repita 2. com LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy = 0.25 e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy = 0.10.

4. Voc\303\252 pode encontrar uma f\303\263mula geral para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy que satisfa\303\247a para qualquer LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy > 0?

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Exemplo 2.5: <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> </Text-field>

1. Find the value of this limit (if it exists).

2. Find values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy that satisfy the conditions of the definition for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy = 0.50, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy = 0.25, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy = 0.10.

Tendo completado estes cinco exemplos, voc\303\252 est\303\241 pronto(a) agora para aprender como usar a defini\303\247\303\243o de limites para provar que um limite tem um dado valor.

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Escrevendo Provas <Equation executable="false" style="2D Math_13" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYmUScmIzk0OTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjktSSNtb0dGJDYuUSgmbWludXM7RicvJSVib2xkR0Y2L0Y4USdub3JtYWxGJy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlQtRiw2JlEnJiM5NDg7RidGNEY3RjpGN0Y6RitGN0Y6">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYmUScmIzk0OTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjktSSNtb0dGJDYuUSgmbWludXM7RicvJSVib2xkR0Y2L0Y4USdub3JtYWxGJy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlQtRiw2JlEnJiM5NDg7RidGNEY3RjpGN0Y6RitGN0Y6</Equation> para Limites </Text-field>

A interpreta\303\247\303\243o gr\303\241fica da defini\303\247\303\243o deve ter dado uma melhor compreens\303\243o dos pap\303\251is de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy e 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

Agora, equipado com esta compreens\303\243o, \303\251 apropriado olhar um pouco para " provas LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUScmIzk0OTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEoJm1pbnVzO0YnRjcvJSZmZW5jZUdGNi8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y2LyUqc3ltbWV0cmljR0Y2LyUobGFyZ2VvcEdGNi8lLm1vdmFibGVsaW1pdHNHRjYvJSdhY2NlbnRHRjYvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZOLUYsNiVRJyYjOTQ4O0YnRjRGN0Y3RitGNw== " de limites.

Exemplos 2.6-9 apresenta provas LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUScmIzk0OTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEoJm1pbnVzO0YnRjcvJSZmZW5jZUdGNi8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y2LyUqc3ltbWV0cmljR0Y2LyUobGFyZ2VvcEdGNi8lLm1vdmFibGVsaW1pdHNHRjYvJSdhY2NlbnRHRjYvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZOLUYsNiVRJyYjOTQ4O0YnRjRGN0Y3RitGNw== para fun\303\247\303\265es cada vez mais complicadas. Como voc\303\252 trabalha ( work ) atrav\303\251s destes exemplos, certifique-se para testar suas f\303\263mulas para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy no EpsilonDelta maplet .

<Text-field bookmark="Example 1" style="Heading 2" layout="Heading 2" spaceabove="7">Exemplo 2.6: Fun\303\247\303\243o Linear</Text-field>

restart;

Considere a afirma\303\247\303\243o de que 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 . (Este limite aparece no Exemplo 2.1.) Para provar que este \303\251 correto entre com

f := 3*x-4; a := 3; L := 5;

Nossa meta (goal) \303\251 mostrar que, para qualquer LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy > 0, \303\251 poss\303\255vel encontra um valor positivo( positive )de of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy com a seguinte propriedade. Em qualquer x para o qual

given := abs( x-a ) < delta: given;

\303\251 v\303\241lido; ent\303\243o

goal := abs( f-L ) < epsilon: goal;

precisa tamb\303\251m ser v\303\241lido.

A prova t\303\255pica( proof )come\303\247a por dar uma olhada no lado esquerdo da meta (goal), isto \303\251

start := lhs( goal ): start;

e tentar identificar modos nos quais estes podem ser relacionados (bounded) por uma constante ( constant ) vezes uma pot\303\252ncia( power ) de 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYoLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnRjIvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJ0Y3LUYsNiVRI2VtRidGL0YyRjctRiw2JVEkdW1hRidGL0YyRjItRiw2I1EhRidGMg==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 factors de .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 . Neste exemplo, \303\251 muito f\303\241cil ver que

q1 := start = 3*abs(x-3): q1;

e ent\303\243o

q2 := rhs(q1) < 3*delta: q2;

Neste ponto, mostramos que

q3 := start < rhs(q2): q3;

Se for poss\303\255vel para escolher LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy tal que

constraint := rhs(q3) < epsilon: constraint;

ent\303\243o a prova ser\303\241 completa. Neste caso, a restri\303\247\303\243o acima \303\251 satisfeita sempre que

isolate( constraint, delta );

Este processo mostra que, dado qualquer LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy > 0,

se 0 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21vR0YkNi1RKSZ2ZXJiYXI7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkotRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHRj8vRjZRJ2l0YWxpY0YnLUYyNi1RKCZtaW51cztGJ0Y1RjhGOy9GPkY6RkBGQkZERkZGSEZLLUkjbW5HRiQ2JFEiM0YnRjVGNUYxRjVGK0Y1 < 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 ent\303\243o 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 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjNGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlUScmIzk0ODtGJy8lJ2l0YWxpY0dGPkY1RjVGK0Y1 < 3 (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y6LUYjNiQtSSNtbkdGJDYkUSIzRidGOkY6LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZILyUpYmV2ZWxsZWRHRjlGK0Y6 ) = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy .

Isto conclui a prova de que o limite de 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 em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiM0YnRj5GPkYrRj4= \303\251 5.

<Text-field bookmark="Example 2" style="Heading 2" layout="Heading 2" spaceabove="7">Exemplo 2.7: Fun\303\247\303\243o Racional em uma Singularidade Remov\303\255vel</Text-field>

Considere a afirma\303\247\303\243o que 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 . Para provar que esta afirma\303\247\303\243o \303\251 correta introduza

f := (x^2-9)/(x-3); a := 3; L := 6;

Nossa meta \303\251 mostrar que , para qualquer LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy > 0, \303\251 poss\303\255vel encontrar um valor positivo de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy para o qual, se LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRidGOQ==satisfaz a inequa\303\247\303\243o

given := abs( x-a ) < delta: given;

ent\303\243o a inequa\303\247\303\243o

goal := abs( f-L ) < epsilon: goal;

\303\251 satisfeita tamb\303\251m. A prova come\303\247a como no Exemplo 1 dando uma olhada no lado esquerdo da meta(goal), a saber, em

start := lhs( goal ): start;

e tentar identificar modos nos quais estes podem ser relacionados (bounded) como no Exemplo 1.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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KC1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGOi8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRidGMS1GLDYlUSNlbUYnLyUnaXRhbGljR0Y6RjVGMS1GLDYlUSR1bWFGJ0ZRRjVGNUYrRjU=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de .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 . Isto precisa de um pouco mais de esfor\303\247o do que no exemplo anterior. Primeiro, note que a condi\303\247\303\243o 0 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21vR0YkNi1RKSZ2ZXJiYXI7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkotRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHRj8vRjZRJ2l0YWxpY0YnLUYyNi1RKCZtaW51cztGJ0Y1RjhGOy9GPkY6RkBGQkZERkZGSEZLLUkjbW5HRiQ2JFEiM0YnRjVGNUYxRjVGK0Y1 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy implica que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVElJm5lO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiM0YnRj5GPkYrRj4= e assim n\303\243o existe problema com divis\303\243o por zero zero. Com este potencial problema exclu\303\255do, encontre um denominador comum ( denominator ) e simplifique, obtendo

q1 := start = simplify( start ) assuming x<>3: q1;

e ent\303\243o

q2 := rhs(q1) < delta: q2;

Neste ponto, mostramos que a inequa\303\247\303\243o

q3 := start < rhs(q2): q3;

\303\251 v\303\241lida para qualquer LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRidGOQ== que satisfa\303\247a 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. Se for poss\303\255vel escolher LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy ent\303\243o

constraint := rhs(q3) < epsilon: constraint;

ent\303\243o a prova ser\303\241 completa. No caso, a restri\303\247\303\243o acima \303\251 satisfeita sempre que

isolate( constraint, delta );

Este processo mostra que , dado qualquer LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy >0,

Isto conclui a prova de que 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.

<Text-field bookmark="Example 3" style="Heading 2" layout="Heading 2" spaceabove="7">Exemplo 2.8: Fun\303\247\303\243o com pot\303\252ncias (Power Function)</Text-field>

Considere a afirma\303\247\303\243o que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KEYrLUYjNipGKy1JJ211bmRlckdGJDYlLUkjbW9HRiQ2LVEkbGltRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dRJXRydWVGJy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjE2NjY2NjdlbUYnLUYjNiYtRiw2JVEieEYnLyUnaXRhbGljR0ZKL0Y7USdpdGFsaWNGJy1GNzYtUScmcmFycjtGJ0Y6Rj1GQEZCRkRGRi9GSUY/RktGTS9GUUZPLUkjbW5HRiQ2JFEiM0YnRjpGOi8lLGFjY2VudHVuZGVyR0Y/RistSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuNGVtRicvJSZkZXB0aEdGZm8vJSpsaW5lYnJlYWtHUSVhdXRvRidGKy1JJm1zcXJ0R0YkNiNGVUYrRjotRjc2LVEiPUYnRjpGPUZARkJGREZGRmluRksvRk5RLDAuMjc3Nzc3OGVtRicvRlFGZnAtRmBwNiNGW29GK0Y6RitGOg== . Para provar que esta afirma\303\247\303\243o \303\251 correta, introduza

f := sqrt(x); a := 3; L := sqrt(3);

Nossa meta \303\251 mostrar que, para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy > 0, \303\251 poss\303\255vel encontrar um valor positivo de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy para o qual, se LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= satisfaz a inequa\303\247\303\243o

given := abs( x-a ) < delta: given;

ent\303\243o a desigualdade( inequality )

goal := abs( f-L ) < epsilon: goal;

\303\251 tambem satisfeita. A prova come\303\247a como no Exemplo 1 e 2 olhando o lado esquerdo da meta (goal), a saber, em

start := lhs( goal ): start;

e tentar identificar caminhos nos quais este pode ser relacionado (bounded) como nos exemplos anteriores.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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KC1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGOi8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRidGMS1GLDYlUSNlbUYnLyUnaXRhbGljR0Y6RjVGMS1GLDYlUSR1bWFGJ0ZRRjVGNUYrRjU=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de .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 . Isto precisa um pouco mais de esfor\303\247o que no exemplos anteriores. Mas, primeiro, tecnicamente \303\251 necess\303\241rio to be addressed . A quantidade 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 \303\251 definida (como um n\303\272mero real) somente quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2J1EieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lJ29wYXF1ZUdGMS8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjhRJ25vcm1hbEYn >= 0. Isto \303\251 onde o parenthetic remark sobre x sendo necess\303\241rio estar no dom\303\255nio da fun\303\247\303\243o. Neste caso, a condi\303\247\303\243o 0 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21vR0YkNi5RKSZ2ZXJiYXI7RicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLUYjNiYtRiw2J1EieEYnLyUnaXRhbGljR0ZCRjUvJSdvcGFxdWVHRkIvRjlRJ2l0YWxpY0YnLUYyNi5RKCZtaW51cztGJ0Y1RjhGO0Y+L0ZBRj1GQ0ZFRkdGSUZLRk4tSSNtbkdGJDYmUSIzRidGNUZXRjhGOEYxRjhGK0Y4 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2J1EnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lJ29wYXF1ZUdRJXRydWVGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGOA== precisa ser acrescentada com LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2J1EieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lJ29wYXF1ZUdGMS8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjhRJ25vcm1hbEYn >= 0 para assegurar que a express\303\243o 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 fa\303\247a sentido.

observe que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEoJm1pbnVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiM0YnRj5GPkYrRj4= = 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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KEYrLUYjNiYtSSNtb0dGJDYtUSIoRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSV0cnVlRicvJSpzZXBhcmF0b3JHUSZmYWxzZUYnLyUpc3RyZXRjaHlHRjwvJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkwtRiM2KEYrLUkmbXNxcnRHRiQ2Iy1GLDYlUSJ4RicvJSdpdGFsaWNHRjwvRjhRJ2l0YWxpY0YnLUY0Ni1RKCZtaW51cztGJ0Y3L0Y7Rj9GPS9GQUY/RkJGREZGRkhGSkZNLUZSNiMtSSNtbkdGJDYkUSIzRidGN0YrRjctRjQ2LVEiKUYnRjdGOkY9RkBGQkZERkZGSEZKL0ZOUSwwLjExMTExMTFlbUYnRjctRjQ2LVExJkludmlzaWJsZVRpbWVzO0YnRjdGaG4vRj5GPEZpbkZCRkRGRkZIRkovRk5RLDAuMzMzMzMzM2VtRictRiM2JkYzLUYjNihGK0ZRLUY0Ni1RIitGJ0Y3RmhuRj1GaW5GQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZjcEZqbkYrRjdGYG9GN0YrRjdGK0Y3 e assim 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 .

Quando valores absolutos ( absolute values )s\303\243o aplicados, o resultado \303\251

q1 := start = abs((x-3))/abs(sqrt(x)+sqrt(3)): q1;

Nosso foco est\303\241 agora na dire\303\247\303\243o do denominador no lado direito da express\303\243o anterior, a saber, em

d1 := denom( rhs(q1) ): d1;

Primeiro, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KEYrLUkmbXNxcnRHRiQ2Iy1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiK0YnL0Y7USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZULUYyNiMtSSNtbkdGJDYkUSIzRidGQUYrRkFGK0ZB \303\251 sempre positivo, assim os valores absolutos n\303\243o s\303\243o necess\303\241rios. O denominador pode ser escrito como

d2 := d1 = op(d1);

E mais, porque LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtc3FydEdGJDYjLUYsNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRidGKy9GOVEnbm9ybWFsRic= > 0, vemos que

d3 := rhs(d2) > sqrt(3): d3;

Tomando os rec\303\255procos ( reciprocals )(e notando que Maple converteu a desigualdade para uma <), vemos que

bound := 1/lhs(d3) > 1/rhs(d3): bound;

Estamos prontos agora para retornar ao quociente total( quotient ), recorde que

q1;

O numerador( numerator ) no lado direito pode ser limitado acima ( bounded above ) por epsilon, o termo remanescente \303\251 limitado pelo argumento anterior. O resultado \303\251

q2 := rhs(q1) < delta*rhs(bound): q2;

Neste ponto, mostramos que

q3 := start < rhs(q2): q3;

Se for poss\303\255vel mostrar que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy tal que

constraint := rhs(q3) < epsilon: constraint;

ent\303\243o a prova ser\303\241 completa. Neste caso, a restri\303\247\303\243o acima \303\251 satisfeita sempre que

isolate( constraint, delta );

se 0 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21vR0YkNi1RKSZ2ZXJiYXI7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkotRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHRj8vRjZRJ2l0YWxpY0YnLUYyNi1RKCZtaW51cztGJ0Y1RjhGOy9GPkY6RkBGQkZERkZGSEZLLUkjbW5HRiQ2JFEiM0YnRjVGNUYxRjVGK0Y1 < 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 (e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= >= 0) ent\303\243o 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 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y6LUYjNiZGKy1JJm1zcXJ0R0YkNiMtSSNtbkdGJDYkUSIzRidGOkYrRjovJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRksvJSliZXZlbGxlZEdGOUYrRjo= < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy .

Isto conclui a prova que 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.

<Text-field bookmark="Example 4" style="Heading 2" layout="Heading 2" spaceabove="7">Exemplo 2.9: Fun\303\247\303\243o Quadr\303\241tica</Text-field>

Considere a afirma\303\247\303\243o de que 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 . (Este limite aparece no Exemplo 2.3.) Para provar que esta afirma\303\247\303\243o \303\251 verdadeira, introduza

f := x^2-3*x+3; a := 3; L := 3;

Nosso gol ser\303\241 mostrar que, para qualquer LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy > 0, \303\251 poss\303\255vel achar um valor positivo de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy para o qual, se x satisfaz a desigualdade

given := abs( x-a ) < delta: given;

ent\303\243o a inequa\303\247\303\243o

goal := abs( f-L ) < epsilon: goal;

\303\251 igualmente satisfeita. A prova come\303\247a como em todos os exemplos anteriores olhando para o lado esquerdo do gol (goal), a saber, para

start := lhs( goal ): start;

O aparecimento de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21vR0YkNi1RKSZ2ZXJiYXI7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkotRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHRj8vRjZRJ2l0YWxpY0YnLUYyNi1RKCZtaW51cztGJ0Y1RjhGOy9GPkY6RkBGQkZERkZGSEZLLUkjbW5HRiQ2JFEiM0YnRjVGNUYxRjVGK0Y1 nesta express\303\243o \303\251 bastante \303\263bvio

q1 := start = abs(x)*abs(x-3): q1;

Aplicando a suposi\303\247\303\243o( assumption ) que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21vR0YkNi1RKSZ2ZXJiYXI7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkotRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHRj8vRjZRJ2l0YWxpY0YnLUYyNi1RKCZtaW51cztGJ0Y1RjhGOy9GPkY6RkBGQkZERkZGSEZLLUkjbW5HRiQ2JFEiM0YnRjVGNUYxRjVGK0Y1 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy leva para

q2 := rhs(q1) < abs(x)*delta: q2;

Resta agora lidar com o termo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21vR0YkNi1RKSZ2ZXJiYXI7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkotRiw2JVEieEYnLyUnaXRhbGljR0Y/L0Y2USdpdGFsaWNGJ0YxRjVGK0Y1 no lado direito desta equa\303\247\303\243o. O enfoque geral \303\251 similar ao que foi tomado no Exemplo 3, mas com um pouco mais de detalhes envolvidos. A chave \303\251 observar que a condi\303\247\303\243o 0 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21vR0YkNi1RKSZ2ZXJiYXI7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkotRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHRj8vRjZRJ2l0YWxpY0YnLUYyNi1RKCZtaW51cztGJ0Y1RjhGOy9GPkY6RkBGQkZERkZGSEZLLUkjbW5HRiQ2JFEiM0YnRjVGNUYxRjVGK0Y1 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy pode ser eq\303\274ivalentemente escrita como

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JS1JI21vR0YkNi1RKCZtaW51cztGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGOi8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSS1GLDYlUScmIzk0ODtGJy8lJ2l0YWxpY0dGOkY1RjVGK0Y1 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEoJm1pbnVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiM0YnRj5GPkYrRj4= < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy (e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVElJm5lO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiM0YnRj5GPkYrRj4= )

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjNGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSgmbWludXM7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLUYsNiVRJyYjOTQ4O0YnLyUnaXRhbGljR0Y+RjVGNUYrRjU= < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUScmIzk0ODtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiK0YnRjcvJSZmZW5jZUdGNi8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y2LyUqc3ltbWV0cmljR0Y2LyUobGFyZ2VvcEdGNi8lLm1vdmFibGVsaW1pdHNHRjYvJSdhY2NlbnRHRjYvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZOLUkjbW5HRiQ2JFEiM0YnRjdGN0YrRjc=

Agora, porque LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSI8RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZNLUYsNiVRJyYjOTQ4O0YnLyUnaXRhbGljR0Y+RjVGNUYrRjU= , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjNGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSIrRidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZNLUYsNiVRJyYjOTQ4O0YnLyUnaXRhbGljR0Y+RjVGNUYrRjU= > LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21vR0YkNi1RKSZ2ZXJiYXI7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkotRiM2Ji1JI21uR0YkNiRRIjNGJ0Y1LUYyNi1RKCZtaW51cztGJ0Y1RjhGOy9GPkY6RkBGQkZERkZGSEZLLUYsNiVRJyYjOTQ4O0YnLyUnaXRhbGljR0Y6RjVGNUYxRjVGK0Y1 e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21vR0YkNi1RKSZ2ZXJiYXI7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkotRiw2JVEieEYnLyUnaXRhbGljR0Y/L0Y2USdpdGFsaWNGJ0YxRjVGK0Y1 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjNGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSIrRidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZNLUYsNiVRJyYjOTQ4O0YnLyUnaXRhbGljR0Y+RjVGNUYrRjU=, assim

q3 := rhs(q2) < (3+delta)*delta: q3;

Se for poss\303\255vel escolher LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy tal que

constraint := rhs(q3) < epsilon: constraint;

ent\303\243o a prova ser\303\241 completa. Neste caso, a restri\303\247\303\243o acima \303\251 satisfeita sempre que

solve( constraint, delta );

Bem, o fato de a resposta do Maple ser vazia significa que o Maple foi incapaz de resolver ( solve ) esta inequa\303\247\303\243o. Se a inequa\303\247\303\243o for convertida em uma equa\303\247\303\243o

c2 := convert( constraint, equality ): c2;

Fica aparente que esta \303\251 um quadr\303\241tica ( quadratic ) em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy (com LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy como um par\303\242metro). As duas raizes s\303\243o

r := solve( c2, delta );

Estas ra\303\255zes s\303\243o reais porque LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy > 0. E mais: uma \303\251 positiva e outra negativa( negative ). N\303\243o \303\251 dif\303\255cil determinar que a restri\303\247\303\243o \303\251 satisfeita para qualquerl LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy com 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 .

Este processo mostra que , dado qualquer LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy >0,

se 0 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21vR0YkNi1RKSZ2ZXJiYXI7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkotRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHRj8vRjZRJ2l0YWxpY0YnLUYyNi1RKCZtaW51cztGJ0Y1RjhGOy9GPkY6RkBGQkZERkZGSEZLLUkjbW5HRiQ2JFEiM0YnRjVGNUYxRjVGK0Y1 < 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 ent\303\243o 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 < 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 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy .

Isto conclui a prova que 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.

QUESTAO 2.1

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZicC1JI21pR0YkNiVRKVByb3ZhbmRvRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEifkYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZMLUYsNiVRI2VtRidGL0YyRjUtRiw2JVEoZXBzaWxvbkYnL0YwRj1GOS1GNjYtUSomdW1pbnVzMDtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GWi1GLDYlUSZkZWx0YUYnRlVGOUY1LUYsNiVRJHF1ZUYnRi9GMkY1LUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjBlbUYnLyUmZGVwdGhHRmFvLyUqbGluZWJyZWFrR1EobmV3bGluZUYnLUZdbzYmRl9vRmJvRmVvL0Zob1ElYXV0b0YnRlxvRjUtRiw2JVElbGltdEYnRi9GMkY1RjUtSShtZmVuY2VkR0YkNiQtRiM2KC1JI21uR0YkNiRRIjRGJ0Y5RjUtRiw2JVEieEYnRi9GMi1GNjYtUSIrRidGOUY7Rj5GQEZCRkRGRkZIRllGZW4tRmdwNiRRIjhGJ0Y5RjlGOS1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORmdxLUZncDYkUSM0MEYnRjlGXG9Gam9GanAtRjY2LVEoJnNyYXJyO0YnRjlGO0Y+RkBGQkZERkZGSEZKRk1GYHFGNUZcb0Zqb0Zcb0Zqby1GLDYlUSVRdWFsRidGL0YyRjUtRiw2JVEnJiMyMzM7RidGL0YyRjUtRiw2JVEiYUYnRi9GMkY1LUYsNiVRLWYmIzI0MztybXVsYUYnRi9GMkY1LUYsNiVRJmdlcmFsRidGL0YyRjUtRiw2JVElcGFyYUYnRi9GMkY1LUYsNiVRIm9GJ0YvRjJGNUZmbkY1RmluRjUtRiw2JVEvc2F0aXNmYSYjMjMxO2FGJ0YvRjJGNUZlckY1LUYsNiVRNGRlZmluaSYjMjMxOyYjMjI3O29GJ0YvRjJGNS1GNjYvUSNkb0YnLyUlYm9sZEdGMS9GM1ElYm9sZEYnLyUrZm9udHdlaWdodEdGYHRGO0Y+RkBGQkZERkZGSEZKRk1GNS1GLDYlUSdsaW1pdGVGJ0YvRjItSSZtdGV4dEdGJDYlUSI/RidGL0YyRlxvRmpvLUYsNiVRJ0RpZ2l0ZUYnRi9GMkY1LUYsNiVRJHN1YUYnRi9GMkY1LUYsNiVRKXJlc3Bvc3RhRidGL0YyRjUtRiw2JVEjbmFGJ0YvRjJGNS1GLDYlUSZmb3JtYUYnRi9GMkY1LUYsNiVRI2RlRidGL0YyRjVGZm5GNUZjcUY1RlJGNS1GNjYtUSIqRidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjE2NjY2NjdlbUYnL0ZORmB2RjUtRiw2JVEqY29uc3RhbnRlRidGL0YyRlxvRmpvLUYsNiNRIUYnLyUrZXhlY3V0YWJsZUdGPUY5

QUESTAO 2.2

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

QUESTAO 2.3

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<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Limites One-Sided (One-Sided Limits)</Text-field>

Para limites two-sided ( two-sided ), valores da vari\303\241vel independente ( independent variable ) nos dois lados do ponto de limite podem ser considerados (supondo que estejam no dom\303\255nio da fun\303\247\303\243o). Limites one-sided somente consideram valores da vari\303\241vel independente em um lado do ponto de limite. Estas diferen\303\247as est\303\243o resumidas na Tabela 2.1.

Tipo de Limite

Valores de Vari\303\241vel Independente

Compact Form

Long Form

two-sided

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one-sided (direita)

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 e 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

one-sided (esquerda)

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 e 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

Tabela 2.1 Distin\303\247\303\265es entre limites two-sided e one-sided

Provas envolvendo limites one-sided seguem o mesmo padr\303\243o geral como de um limite two-sided. A principal diferen\303\247a \303\251 que \303\251 poss\303\255vel usar a propriedade adicional LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= > LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= (limite \303\240 direita) ou LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= (limite \303\240 esquerda).

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

Quando usamos a defini\303\247\303\243o de limite para mostrar 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 , uma imagem \303\272til poderia conter as seguintes caracter\303\255sticas

o gr\303\241fico( graph )da fun\303\247\303\243o pr\303\263ximo ao ponto de limite, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRImFGJ0Y0RjdGPkYrRj4= ,

uma faixa horizontal centrada em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRIkxGJ0Y0RjdGPkYrRj4= com 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 ,

uma faixa vertical centrada em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRImFGJ0Y0RjdGPkYrRj4= com 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 (tecnicamente, a linha vertical LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRImFGJ0Y0RjdGPkYrRj4= poderia ser exclu\303\255da desta faixa)

Se o valor de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy \303\251 consistente com a defini\303\247\303\243o do limite, ent\303\243o cada valor de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= na banda vertical(exceto, possivelmente, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRImFGJ0Y0RjdGPkYrRj4= ) produzir\303\241 um valor de fun\303\247\303\243o que est\303\241 contido na faixa horizontal. Se existir um \303\272nico( even )valor de fun\303\247\303\243o que esteja fora da faixa horizontal, ent\303\243o o valor de of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy ser\303\241 demasiado grande para este LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy ou o valor do limite (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= ) est\303\241 incorreto.

Provas de limites s\303\243o as mais f\303\241ceis de completar quando \303\251 poss\303\255vel escrever 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 como um produto( product ) de um pot\303\252ncia de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21vR0YkNi1RKSZ2ZXJiYXI7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkotRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHRj8vRjZRJ2l0YWxpY0YnLUYyNi1RKCZtaW51cztGJ0Y1RjhGOy9GPkY6RkBGQkZERkZGSEZLLUYsNiVRImFGJ0ZSRlRGNUYxRjVGK0Y1 e uma express\303\243o com as restri\303\247\303\265es .

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">O pr\303\263ximo qual \303\251?</Text-field>

O pr\303\263ximo cap\303\255tulo desenvolve uma cole\303\247\303\243o de leis de limite que podem ser usadas ao inv\303\251s de uma prova LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUScmIzk0OTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEoJm1pbnVzO0YnRjcvJSZmZW5jZUdGNi8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y2LyUqc3ltbWV0cmljR0Y2LyUobGFyZ2VvcEdGNi8lLm1vdmFibGVsaW1pdHNHRjYvJSdhY2NlbnRHRjYvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZOLUYsNiVRJyYjOTQ4O0YnRjRGN0Y3RitGNw== .

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