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de Estudo de C\303\241lculo\302\251 Maplesoft, a division of Waterloo Maple Inc., 2005Cap\303\255tulo 5: Limites no Infinito
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Introdu\303\247\303\243o</Font></Text-field>Quando uma vari\303\241vel independente ( independent variable ) tal qual LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= cresce sem limites ( bound), isto \303\251, torna-se maior que qualquer n\303\272mero finito ( finite ) que se possa conceber, n\303\263s dizemos que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= tende ao infinito e escrevemos LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEnJnJhcnI7RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtRiw2JVEoJiM4NzM0O0YnRjRGN0Y+RitGPg==Da mesma forma, se uma vari\303\241vel independente como x decresce sem limites (bound), isto \303\251, torna-se menor que qualquer n\303\272mero finito que se possa conceber, dizemos que x tende a menos infinito e escrevemos LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEnJnJhcnI7RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtRiM2JS1GOzYtUSgmbWludXM7RidGPkZARkNGRUZHRklGS0ZNRk9GUi1GLDYlUSgmIzg3MzQ7RidGNEY3Rj5GK0Y+RitGPg==Um limite no infinito ( infinity ) \303\251 um limite ( limit ) no qual a vari\303\241vel independente cresce sem limites (bound), e escrevemos um enunciado para limite da forma 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 mesma forma, um limite em menos infinito \303\251 um limite no qual a vari\303\241vel independente decresce sem limtes (bound), e escrevemos a declara\303\247\303\243o de limite da 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Quando escrevemos uma nota\303\247\303\243o como 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, estamos tentando dizer que o ponto de limite \303\251 o infinito ( infinity). Mas infinito n\303\243o \303\251 um n\303\272mero real ( real number ) e tais limites requerem tratamento especial em termos da defini\303\247\303\243o exata. N\303\243o vamos, contudo, dedicar tempo algum \303\240 esta defini\303\247\303\243o epsilon-delta de limites no infinito. Ao inv\303\251s , vamos nos concentrar em desenvolver uma compreens\303\243o intuitiva de limites no infinito e uma habilidade de manipula\303\247\303\243o necess\303\241ria para avaliar ( evaluate )estes limites. Neste cap\303\255tulo aprenderemos umas poucas t\303\251cnicas para avalia\303\247\303\243o de limites no infinito sem o uso de um gr\303\241fico ( graph). Apesar dos coment\303\241rios anteriores sobre as dificuldades de vizualizar um limite no infinito, n\303\263s veremos como gr\303\241ficos ( graphs ) podem ser usados para ajudar com a avalia\303\247\303\243o de um limite no infinito. O cap\303\255tilo Limites no Infinito continua alguns dos t\303\263picos introduzidos aqui.
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Cap\303\255tulo Gloss\303\241rio</Font></Text-field>Os seguintes termos, neste cap\303\255tulo, t\303\252m conex\303\243o com o Dicion\303\241rio de Matem\303\241tica Maple ( Maple Math Dictionary).
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Initializa\303\247\303\243o</Font></Text-field>restart;
with( plots ):
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font opaque="true" background="[255,255,0]">Reciprocal </Font><Hyperlink linktarget="Dictionary:Definition,exponent" hyperlink="true"><Font style="Heading 1" opaque="true" background="[255,255,0]">Powers</Font></Hyperlink><Font encoding="UTF-8"> e Fun\303\247\303\265es Racionais</Font></Text-field>O fato fundamental usado para avaliar todos os limites no infinito para uma fun\303\247\303\243o racional ( rationalfunction ) \303\251 o enunciado bastante \303\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 = 0 O Produto, Leis de Pot\303\252ncia ( Power ) e Raiz ( Root ) podem ser usadas para achar o limite no infinito de qualquer pot\303\252ncia (negativa). Para todo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI+RidGOUY7Rj5GQEZCRkRGRkZIRkpGTUY1LUkjbW5HRiQ2JFEiMEYnRjktRjY2LVEiLEYnRjlGO0Y+RkBGQkZERkZGSEZKRk1GOQ==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 = 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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0YvSe LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJrRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4= , ent\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjgtRiM2JkYrLUYjNiZGKy1JJW1zdXBHRiQ2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjlRJ2l0YWxpY0YnLUY1NiRRIjBGJ0Y4LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YrRjhGK0Y4LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZVLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRitGOA== = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUklbXN1cEdGJDYlLUYsNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtbkdGJDYkUSIwRicvRjtRJ25vcm1hbEYnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YrRkFGK0ZB = 1 para todo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVElJm5lO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4= e assim 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 = 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 = 1 O limite no infinito para pot\303\252ncias positivas ( positive ) de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= ser\303\241 discutido no pr\303\263ximo cap\303\255tulo.Verifica\303\247\303\243o gr\303\241fica destes resultados n\303\243o \303\251 realmente necess\303\241ria. Mas, \303\251 importante que voc\303\252 entenda bem que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjgtRiM2JkYrLUklbXN1cEdGJDYlLUYsNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GOVEnaXRhbGljRictRiw2JVEia0YnRkNGRi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGK0Y4LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZTLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRitGOA== decresce at\303\251 zero ( zero ) mais rapidamente que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjgtRiM2JkYrLUklbXN1cEdGJDYlLUYsNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GOVEnaXRhbGljRictRiw2JVEibUYnRkNGRi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGK0Y4LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZTLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRitGOA== quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= >LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= >0. Este ponto est\303\241 ilustrado na Figura 5.1.
O limite no infinito de uma fun\303\247\303\243o racional em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 tipicamente avaliada pela identifica\303\247\303\243o da mais alta pot\303\252ncia de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= no denominador ( denominator ) e dividindo ambos numerador ( numerator ) e denominador por esta pot\303\252ncia de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= . A express\303\243o obtida desta manipula\303\247\303\243o ter\303\241 um denominador cujo limite no infinito existe e n\303\243o \303\251 zero. Se o numerador n\303\243o cont\303\251m pot\303\252ncias positivas de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= , ent\303\243o a Regra do Quociente ( Quotient Rule ) pode ser usada para determinar o valor deste limite no infinito. O caso com pot\303\252ncias positivas de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= no numerador ser\303\241 discutida no Cap\303\255tulo 6, o cap\303\255tulo de Limites Infinitos ( Infinite Limits ).
<Text-field style="Heading 2" layout="Heading 2">Exemplo 5.1</Text-field>Para o nosso primeiro Exemplo, considere o limite no infinito de uma fun\303\247\303\243o racional, na qual o numerador e o denominador sejam ambos polin\303\264mios de quinto grau ( quinticpolynomials ). Tal fun\303\247\303\243o poderia serf1 := (63*x^5-39*x^4-48*x^3-94*x^2+10*x-20)/(70*x^5+98*x^4+58*x^3+45*x^2-12*x+28):
n1 := numer( f1 ):
d1 := denom( f1 ):
L1 := Limit( f1, x=infinity ):
L1;O mais alto grau ( degree ) no denominador \303\251k := degree(d1);Dividindo ambos numerador e denominador por 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 produz -se o problema equivalenten2 := expand( n1/x^k ):
d2 := expand( d1/x^k ):
f2 := n2/d2:
L2 := Limit( f2, x=infinity ):
L1 = L2;Note que o limite do denominador existe e n\303\243o \303\251 zero, como se v\303\252 no Maple viad3 := Limit( d2, x=infinity ):
d3 = value(d3);A Regra do Quociente para limites pode ser aplicada para determinar que o valor para este limite \303\251 dado porL3 := Limit( n2, x=infinity ) / d3:
L2 = L3;
`` = value( L3 );Figura 5.2, contendo um gr\303\241fico desta fun\303\247\303\243o, confirma que este \303\251 um resultado razo\303\241vel.
plot( f1, x=1..200, discont=true );
Figura 5.2Um gr\303\241fico da fun\303\247\303\243o 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
Note, entretanto, que o gr\303\241fico \303\251 insuficiente sem a manipula\303\247\303\243o alg\303\251brica. \303\211 poss\303\255vel que a fun\303\247\303\243o pare\303\247a bem comportada no intervalo ( interval ) finito mas tenha comportamento diferente para valores mais altos de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= .
<Text-field style="Heading 2" layout="Heading 2">Exemplo 5.2</Text-field>Considere o limite no infinito de uma fun\303\247\303\243o racional cujo denominador tenha um grau mais alto que o numerador . Tal fun\303\247\303\243o poderia serf1 := (93*x^6+45*x^5+81*x^4+8*x^2-44*x-80)/(63*x^9-24*x^8-63*x^7-36*x^5+85*x^3+35):
n1 := numer( f1 ):
d1 := denom( f1 ):
L1 := Limit( f1, x=infinity ):
L1;O mais alto grau no denominador \303\251k := degree(d1);Dividindo ambos numerados e dnominador por 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 produz -se o problema equivalenten2 := expand( n1/x^k ):
d2 := expand( d1/x^k ):
f2 := n2/d2:
L2 := Limit( f2, x=infinity ):
L1 = L2;Note que este limite do denominador existe e \303\251 diferente de zero, como se v\303\252 no Maple viad3 := Limit( d2, x=infinity ):
d3 = value(d3);A Regra do Quociente para limites pode ser aplicada para determinar que o valor para este limite \303\251 dado porL3 := Limit( n2, x=infinity ) / d3:
L2 = L3;
`` = value( L3 );Figura 5.3, contendo um gr\303\241fico desta fun\303\247\303\243o, confirma (a razoabilidade???) deste resultado.
plot( f1, x=1..10, discont=true );
Figura 5.3 Um gr\303\241fico da fun\303\247\303\243o 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
Note, entretanto, que o gr\303\241fico \303\251 insuficiente sem a manipula\303\247\303\243o alg\303\251brica. \303\211 poss\303\255vel que a fun\303\247\303\243o pare\303\247a bem comportada no intervalo ( interval ) finito mas tenha comportamento diferente para valores mais altos de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= .Quest\303\243o 5.1
Encontre o seguinte limite: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYwLUkjbWlHRiQ2JVElbGltdEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY1RjUtSSZtZnJhY0dGJDYoLUkobWZlbmNlZEdGJDYkLUYjNipGNS1JI21uR0YkNiRRIzE0RidGOUY1LUYsNiVRInhGJ0YvRjItRjY2LVEiK0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZcby1GWDYkUSMxMUYnRjkvJStleGVjdXRhYmxlR0Y9RjlGOS1GIzYpLUZYNiRRIzEwRidGOUY1LUklbXN1cEdGJDYlRmVuLUYjNiUtRlg2JFEiMkYnRjlGL0YyLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Zobi1GWDYkUSIxRidGOUYvRjIvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRltxLyUpYmV2ZWxsZWRHRj0tRiw2I1EhRictSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGaHEvJSpsaW5lYnJlYWtHUShuZXdsaW5lRidGNUZlbi1GNjYtUSgmc3JhcnI7RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS1GNjYtUS8mUHJvcG9ydGlvbmFsO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yNzc3Nzc4ZW1GJy9GTkZockY1RmFvRjk=Se o limite \303\251 indefinido, digite sua resposta como dne. Se o limite for infinito, digite sua resposta como infinito.
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Fun\303\247\303\265es Peri\303\263dicas</Font></Text-field>As fun\303\247\303\265es 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 e 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 s\303\243o peri\303\263dicas ( periodic ) com per\303\255odo ( 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 . Isto significa que qualquer valor entre -1 e 1 (inclusive) \303\251 obtido para valores arbitrariamnete altos de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy . Portanto, seus limites no infinito n\303\243o existem (n.e.), e 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 = n.e.
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 = n.e. Figura 5.4 sugere que este poderia ser o caso.
<Text-field style="Heading 2" layout="Heading 2">Exemplo 5.3</Text-field>Considere o limite no infinito de um produto ( product ) entre uma fun\303\247\303\243o peri\303\263dica e uma fun\303\247\303\243o racional. Tal produto poderia ser o seguinte.f1 := sin(x)*(x-1)/(x^2+x+1):
L1 := Limit( f1, x=infinity ):
L1;A fun\303\247\303\243o neste exemplo \303\251 o produto de uma fun\303\247\303\243o peri\303\263dica e uma express\303\243o racional. A fun\303\247\303\243o racional neste exemplo se v\303\252, aproxima-se de zero quando x cresce sem limites ( bound). A Regra do Produto ( Product Rule ) n\303\243o pode ser usada para concluir que o liite \303\251 zero porque 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 n\303\243o existe. Em situa\303\247\303\265es como estas o Teorema do Confronto ( Squeeze Theorem ) pode ser aplicado. Porque a fun\303\247\303\243o peri\303\263dica neste problema \303\251 limitada (bounded) (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) para todo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e a fun\303\247\303\243o racional \303\251 positiva para todo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2J1EieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lJ29wYXF1ZUdGMS8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjhRJ25vcm1hbEYn >1, Podemos concluir que para todo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkktRjY2LVEiPkYnRjJGOUY7Rj1GP0ZBRkNGRUZHRkpGNS1JI21uR0YkNiRRIjFGJ0YyLUY2Ni1RIixGJ0YyRjlGO0Y9Rj9GQUZDRkVGR0ZKRjI=
Confirma\303\247\303\243o gr\303\241fica do resultado \303\251 fornecida na Figura 5.5.
g1 := (x-1)/(x^2+x+1):
f1 := sin(x)*g1:
plot( [f1, abs(g1),-abs(g1)], x=10..200,
numpoints=200, color=[red,blue,cyan] );Figura 5.5 Confirma\303\247\303\243o gr\303\241fica da inequa\303\247\303\243o em (5.1)
Agora, porqueL2 := Limit( g1, x=infinity ):
L2 = value( L2 );eL3 := Limit( -g1, x=infinity ):
L3 = value( L3 );O Teorema do Confronto nos diz queL1 = value( L2 );Quest\303\243o 5.2
Encontre o seguinte limite: 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Se o limite \303\251 indefinido, digite sua resposta como dne. Se o limite for infinito, digite sua resposta como infinito.
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Limites em Menos Infinito</Text-field>Limites em menos infinito pode ser convertido em limites no infinito pela simples troca de vari\303\241veis ( variables )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The limits 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pode ser avaliado somente quando 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 \303\251 definido. Isto significa que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= deve ser ou um n\303\272mero inteiro ou um n\303\272mero racional com um denominador impar ( odd ). O gr\303\241fico de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjgtRiM2JkYrLUklbXN1cEdGJDYlLUYsNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GOVEnaXRhbGljRictRiw2JVEia0YnRkNGRi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGK0Y4LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZTLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRitGOA== para valores negativos de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 intimamente relacionado com o gr\303\241fico correspondente para valores positivos LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, mas a exata rela\303\247\303\243o depende se LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLDYlUSJrRidGNUY4LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YrL0Y5USdub3JtYWxGJw== \303\251 uma fun\303\247\303\243o par ou \303\255mpar.Figura 5.6 mostra gr\303\241fico de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjgtRiM2JkYrLUklbXN1cEdGJDYlLUYsNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GOVEnaXRhbGljRictRiw2JVEia0YnRkNGRi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGK0Y4LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZTLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRitGOA== para v\303\241rios valores positivos de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4=.
Portanto, para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJrRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4=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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkctRjM2LVExJkludmlzaWJsZVRpbWVzO0YnRi9GNkY5RjtGPUY/RkFGQ0ZFRkhGLw==e para todos os valores racionais positivos de 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 (na forma reduzida) com LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\255mpar, 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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEjMC5GJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGLw==J\303\241 que limites em menos infinito s\303\243o facilmente convertidos em limite no infinito, as t\303\251cnicas desenvolvidas para limites no infinito podem tambem ser aplicadas para limites em menos infinito.
<Text-field style="Heading 2" layout="Heading 2">Exemplo 5.4</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font family="Arial">Note</Font></Text-field>Except that infinity is changed to -infinity, this discussion is exactly the same as the one in Example 5.1.Para o nosso quarto exemplo, considere o limite em meno infinito para uma fun\303\247\303\243o racional cujo o numerador e denominador numerator s\303\243o ambos polin\303\264mios de quinto grau. Tal fun\303\247\303\243o \303\251 f1 := (63*x^5-39*x^4-48*x^3-94*x^2+10*x-20)/(70*x^5+98*x^4+58*x^3+45*x^2-12*x+28):
n1 := numer( f1 ):
d1 := denom( f1 ):
L1 := Limit( f1, x=-infinity ):
L1;O mais alto grau no denominador \303\251k := degree(d1);Dividindo ambos numerador e denominador por 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 produz -se o problema equivalenten2 := expand( n1/x^k ):
d2 := expand( d1/x^k ):
f2 := n2/d2:
L2 := Limit( f2, x=-infinity ):
L1 = L2;Note que o limite do denominador existe e n\303\243o \303\251 zero, como se v\303\252 no Maple viad3 := Limit( d2, x=-infinity ):
d3 = value(d3);A Regra do Quociente para limites pode ser aplicada para determinar que o valor para este limite \303\251 L3 := Limit( n2, x=-infinity ) / d3:
L2 = L3;
`` = value( L3 );Figura 5.7 contendo o gr\303\241fico da dada fun\303\247\303\243o racional confirma de forma razo\303\241vel este resultado rationalfunction confirms.
plot( f1, x=-1000..-10, discont=true );
Figura 5.7 Gr\303\241fico da fun\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ky1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RidGNEY3Rj5GPi1GOzYtUTEmSW52aXNpYmxlVGltZXM7RidGPkZARkNGRUZHRklGS0ZNRk9GUi1GOzYtUSI9RidGPkZARkNGRUZHRklGS0ZNRk9GUkZmbi1JJm1mcmFjR0YkNiktRiM2JkYrLUYjNiZGKy1GIzYmRistRiM2L0YrLUYjNictSSNtbkdGJDYlUSM2M0YnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Y+RmZuLUYjNiZGKy1JJW1zdXBHRiQ2JS1GLDYmRmVuRjRGXXBGNy1Gam82JVEiNUYnRl1wRj4vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRitGPkYrRj4tRjs2LlEoJm1pbnVzO0YnRl1wRj5GQC9GREY2RkVGR0ZJRktGTUZPL0ZTUSwwLjMzMzMzMzNlbUYnLUYjNictRmpvNiVRIzM5RidGXXBGPkZmbi1GIzYmRistRmNwNiVGZXAtRmpvNiVRIjRGJ0ZdcEY+RmpwRitGPkYrRj5GXXEtRiM2Jy1Gam82JVEjNDhGJ0ZdcEY+RmZuLUYjNiZGKy1GY3A2JUZlcC1Gam82JVEiM0YnRl1wRj5GanBGK0Y+RitGPkZdcS1GIzYnLUZqbzYlUSM5NEYnRl1wRj5GZm4tRiM2JkYrLUZjcDYlRmVwLUZqbzYlUSIyRidGXXBGPkZqcEYrRj5GK0Y+LUY7Ni5RIitGJ0ZdcEY+RkBGQ0ZFRkdGSUZLRk0vRlBRLDAuMjIyMjIyMmVtRicvRlNGW3QtRiM2Ji1Gam82JVEjMTBGJ0ZdcEY+RmZuRmVwRj5GXXEtRmpvNiVRIzIwRidGXXBGPkY+RitGPkYrRj5GK0Y+LUYjNiZGKy1GIzYmRistRiM2JkYrLUYjNi9GKy1GIzYnLUZqbzYlUSM3MEYnRl1wRj5GZm5GYHBGK0Y+RmdzLUYjNictRmpvNiVRIzk4RidGXXBGPkZmbkZocUYrRj5GZ3MtRiM2Jy1Gam82JVEjNThGJ0ZdcEY+RmZuRmRyRitGPkZncy1GIzYnLUZqbzYlUSM0NUYnRl1wRj5GZm5GYHNGK0Y+Rl1xLUYjNiYtRmpvNiVRIzEyRidGXXBGPkZmbkZlcEY+RmdzLUZqbzYlUSMyOEYnRl1wRj5GPkYrRj5GK0Y+RitGPi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGXncvJSliZXZlbGxlZEdGQi8lK2JhY2tncm91bmRHRl9wRitGPkYrRj4=
Note, entretanto, que o gr\303\241fico \303\251 insuficiente sem a manipula\303\247\303\243o alg\303\251brica. \303\211 poss\303\255vel que a fun\303\247\303\243o pare\303\247a bem comportada no intervalo ( interval ) finito mas tenha comportamento diferente para valores mais altos de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= .Quest\303\243o 5.3
Encontre o seguinte limite: 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 o limite \303\251 indefinido, digite sua resposta como dne. Se o limite for infinito, digite sua resposta como infinito.
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Ass\303\255ntota Horizontal </Font></Text-field>
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">Defini\303\247\303\243o - Ass\303\255ntota Horizontal </Font></Text-field>O gr\303\241fico de 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 \303\251 dito ter um aass\303\255ntota ( asymptote ) horizontal se, para a linha(horizontal) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUklbXN1YkdGJDYlLUYsNiVRImhGJ0Y0RjctRiM2JEZXRj4vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0YrRj5GK0Y+, pelo menos um dos limites 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 ou 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 f\303\264r mantido.
<Text-field style="Heading 2" layout="Heading 2">Exemplo 5.5</Text-field>Encontre a(s) ass\303\255ntota(s) horizontal(is) para 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 , que \303\251, paraf := (3*x^5+1)/(abs(x)^5+5*x^2+1) + 10*sin(x)/(x^2+1);
Enquanto o primeiro termo nesta fun\303\247\303\243o n\303\243o \303\251 uma fun\303\247\303\243o racional, ela pode ser tratada exatamente da mesma forma. Limites para o segundo termo podem ser avaliados com a ajuda do Teorema do Comfronto(como no exemplo 3).Observe que para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= > 0, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNiYtSSNtb0dGJDYtUSkmdmVyYmFyO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPC8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y8LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjwvJSdhY2NlbnRHRjwvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZMLUYsNiVRInhGJy8lJ2l0YWxpY0dGQS9GOFEnaXRhbGljRidGM0Y3LUY0Ni1RIj1GJ0Y3RjpGPS9GQEY8RkJGREZGRkgvRktRLDAuMjc3Nzc3OGVtRicvRk5GZW5GT0Y3RitGNw== e assim 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 . O limite no infinito desta fun\303\247\303\243o v\303\252-se facilmente que \303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEiM0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv. Por exemplo, Maple d\303\241 y_h1 := limit( f, x=infinity );Da mesma forma, para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= < 0, 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 e assim 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 . O limite em menos infinito desta fun\303\247\303\243o v\303\252-se facilmente que \303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JS1JI21vR0YkNi1RKCZtaW51cztGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGOi8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjNGJ0Y1RjVGK0Y1. Por exemplo, Maple d\303\241 y_h2 := limit( f, x=-infinity );A verifica\303\247\303\243o gr\303\241fica destes resultados pode ser vista na Figura 5.8.y_h2 := limit( f, x=-infinity );A verifica\303\247\303\243o destes resultados podem ser vistos na Figura 5.8.
plot([ f, y_h1, y_h2 ], x=-20..20, y=-5..5, discont=true,
color=[red,blue,green], thickness=3,
legend =["y = f(x)", "y = 3", "y = -3"]);Figura 5.8 Gr\303\241fico da fun\303\247\303\243o 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
Note como a fun\303\247\303\243o oscila mas se estabelesce em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiM0YnRj5GPkYrRj4= para altos valores positivos de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, e em 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 para altos valores negativos de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2J1EieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lJ29wYXF1ZUdGMS8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjhRJ25vcm1hbEYn .Quest\303\243o 5.4
Encontre todas as ass\303\255ntotas horizontais de: 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Identificar cada ass\303\255ntota como equa\303\247\303\243o y = L para um valor apropriado de L, digite sua resposta na forma{y=L)} ou {y = L1, y=L2}.
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Cap\303\255tulo Resumo</Font></Text-field>Uma das mais importantes ferramentas na avalia\303\247\303\243o de limites no infinito (ou em menos infinito) \303\251 o limite fundamental 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Este fato \303\251 utilizado para avaliar limites no infinito de fun\303\247\303\265es racionais. Uma das aplica\303\247\303\265es fundamentais de limite no infinito \303\251 encontrar a ass\303\255ntota(s) horizontal(s) da fun\303\247\303\243o.
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Qual \303\251 o Pr\303\263ximo?</Font></Text-field>O pr\303\263ximo cap\303\255tulo continua a discuss\303\243o de limites e infinito. Dois dos t\303\263picos de discuss\303\243o nesse cap\303\255tulo ser\303\241 (1) limites no infinito de polin\303\264mios racionais onde o grau do numerador \303\251 maior que o do denominador; e (2), Ass\303\255ntotas verticais e obl\303\255quas (oblique ).Chapter Navigation<< Prev1234 5 678910111213141516171819202122232425262728293031Next >>
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