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de estudo de c\303\241lculo\302\251 Maplesoft, a division of Waterloo Maple Inc., 2005Cap\303\255tulo 4: Limites para Fun\303\247\303\265es Trigonom\303\251tricas
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Introdu\303\247\303\243o</Font></Text-field>Limites de express\303\265es envolvendo fun\303\247\303\265es trigonom\303\251tricas ( trigonometricfunctions ) usualmente podem ser avaliadas ( evaluated )por substitui\303\247\303\243o ( substitution ) direta--- a n\303\243o ser que divis\303\243o por zero ( zero ) seja encontrada ou seja encontrado um caso de oscila\303\247\303\243o infinita . Quando divis\303\243o por zero( zero ) for encontrada, pode-se tentar simplifica\303\247\303\265es como as apresentadas no Cap\303\255tulo 1. Em muitos casos, entretanto, isto n\303\243o \303\251 suficiete para avaliar ( evaluate ) um limite ( limit ). Os dois limites 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 and 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 introduzidos -- graficamente -- no Cap\303\255tulo 1. No corrente cap\303\255tulo, cada um destes limites ser\303\241 rigorosamente avaliado. Estas provas ser\303\243o nossa primeira aplica\303\247\303\243o real das leis de limites, particularmente o Teorema do Confronto (Squeeze Theorem).
<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">Inicializa\303\247\303\243o</Font></Text-field>restart;
with( plots ):
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Procedimentos Auxiliares</Text-field>figure := proc( theta )
local A, C, P, S, T, Title, t;
C := implicitplot( x^2+y^2=1, x=-1..1, y=-1..1, tickmarks=[2,2] ):
S := theta -> plot( [[0,0],[cos(theta),sin(theta)],[cos(theta),0]], color=blue ):
A := theta -> plot( [cos(theta)*cos(t),cos(theta)*sin(t),t=0..theta], color=green ):
P := theta -> pointplot( [ [1,0],
[cos(theta),0],
[cos(theta)*cos(theta),cos(theta)*sin(theta)],
[cos(theta),sin(theta)] ],
symbol=circle, symbolsize=12 ):
T := theta -> textplot( {
[-0.1,-0.1,"O"],
[1.25,-0.1,"A (1,0)"],
[cos(theta),-0.1,"B"],
[cos(theta)*cos(theta)-0.1,cos(theta)*sin(theta)+0.1,"C"],
[cos(theta)+0.25,sin(theta)+0.1,"P (cos t, sin t)"],
[-0.2,1.1,"(0,1)"],
[0.3,0.1,"t"] } ):
Title := proc( theta )
if nargs > 1
then
if args[2]=`shorttitle`
then return sprintf( "Unit circle with theta=%07.5f", theta )
else return ""
end if
else
return sprintf( " When t=%07.5f:
area( sector(COB) ) = %07.5f
area( triangle(POB) ) = %07.5f
area( sector(POA) ) = %07.5f",
theta,
abs(theta)*cos(theta)^2/2,
cos(theta)*abs(sin(theta))/2,
abs(theta)/2 ):
end if
end proc;
plots[display]( [C,S(theta),A(theta),P(theta),T(theta)],
view=[-1.7..1.7,-1.7..1.7],
title=Title(args),
scaling=constrained ):
end proc:
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Leis de Limites para Fun\303\247\303\265es Trigonom\303\251tricas</Font></Text-field>As fun\303\247\303\265es de seno ( sine ) e cosseno ( cosine ) s\303\243o as funda\303\247\303\265es de todas as fun\303\247\303\265es trigonom\303\251tricas. Estas fun\303\247\303\265es s\303\243o peri\303\263dicas, com per\303\255odo ( 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 . Da Figura 4.1, o gr\303\241fico (graphs) destas duas fun\303\247\303\265es,
Psin := plot([sin(x),cos(x)], x=-3*Pi..3*Pi,
color=[red,blue],legend=["y = sin(x)", "y = cos(x)"]):
Psin;Figura 4.1 Tr\303\252s per\303\255odos das fun\303\247\303\265es de seno e cosseno
Tab 4.1 Limites b\303\241sicos para fun\303\247\303\265es de seno e cosseno
Os limites b\303\241sicos para outras quatro fun\303\247\303\265es trigonom\303\251tricas, tangente, cotangente, secante e cossecante, s\303\243o obtidos da Tabela 4.1 e do comportamento das fun\303\247\303\265es de seno e cosseno. Baseado nas Leis de Limite, particularmente aquela para quocientes, \303\251 f\303\241cil ver que muitos limites envolvendo estas fun\303\247\303\265es podem ser avaliados por substitui\303\247\303\243o. Por exemplo, Tabela 4.2 lista os mais \303\263bvios destes limites.
Tab 4.2 O limites "\303\263bvios" de algumas fun\303\247\303\265es trigonom\303\251tricas
Estas f\303\263rmulas podem ser resumidas dizendo-se que limites envolvendo quaisquer destas seis fun\303\247\303\265es trigonom\303\251tricas podem ser avaliados por substitui\303\247\303\243o, desde que o ponto de limite esteja no dom\303\255nio ( domain ) da fun\303\247\303\243o. Para as fun\303\247\303\265es tangente ( tangent ) e secante( secant ), pontos onde a fun\303\247\303\243o cosseno \303\251 zero (( multiples ) m\303\272ltiplos \303\255mpares de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEnJiM5NjA7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y6LUYjNiQtSSNtbkdGJDYkUSIyRidGOkY6LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZILyUpYmV2ZWxsZWRHRjlGK0Y6) devem ser exclu\303\255dos; para as fun\303\247\303\265es cotangente ( cotangent ) e cossecante ( cosecant ), os " pontos ruins" s\303\243o os zeros da fun\303\247\303\243o seno, a saber, qualquer (inteiro) m\303\272ltiplo de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjA7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy . Figuras 4.2 - 4.4, criadas com `Figure 4.2` := Psin:
`Figure 4.3` := plot([tan(x),cot(x)], x=-3*Pi..3*Pi, color=[red,blue],
view=[-3*Pi..3*Pi,-10..10], discont=true, legend=["y = tan(x)", "y = cot(x)"]):
`Figure 4.4` := plot([sec(x),csc(x)], x=-3*Pi..3*Pi, color=[red,blue],
view=[-3*Pi..3*Pi,-10..10], discont=true, legend=["y=sec(x)", "y=csc(x)"]):e apresentadas com
`Figure 4.2`;`Figure 4.3`;`Figure 4.4`;
Figura 4.2 Tr\303\252s per\303\255odos das fun\303\247\303\265es seno e cossine
Figure 4.3 Seis per\303\255odos das fun\303\247\303\265es tangente e cotangente
Figure 4.4 Tr\303\252s per\303\255odos das fun\303\247\303\265es secante e cossecante
fornecendo os gr\303\241ficos destas seis fun\303\247\303\265es, pode ser \303\272til para relembrar os dom\303\255nios destas fun\303\247\303\265esO Limit maplet pode ser usado para avaliar limites envolvendo fun\303\247\303\265es trigonom\303\251tricos. Para aplicar as leis, clique no bot\303\243o para a fun\303\247\303\243o ou pegue a fun\303\247\303\243o de uma das caixas que se abrem.
<Text-field style="Heading 1" layout="Heading 2" spaceabove="7"><Font encoding="UTF-8">Uma Rigorosa Avalia\303\247\303\243o de Dois Limites Especiais </Font></Text-field>No cap\303\255tulo 1, argumentos gr\303\241ficos s\303\243o usados para justificar os 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 and 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 os argumentos baseados nos gr\303\241ficos de 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 e 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 sejam indicativos, eles n\303\243o s\303\243o provas.Para fornecer provas rigorosas de que estes limites existem e t\303\252m os valores ( values ) sugeridos pelos gr\303\241ficos, as Leis de Limites, particularmente o Teorema do Confronto (Squeeze Theorem), ser\303\243o utilizadas.
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Prova de que <Equation executable="false" style="2D Math" input-equation="" display="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">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNipGKy1JJ211bmRlckdGJDYlLUkjbW9HRiQ2LVEkbGltRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dRJXRydWVGJy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjE2NjY2NjdlbUYnLUYjNiYtRiw2JVEnJiM5NTI7RicvJSdpdGFsaWNHRj9GOi1GNzYtUScmcmFycjtGJ0Y6Rj1GQEZCRkRGRi9GSUY/RktGTS9GUUZPLUkjbW5HRiQ2JFEiMEYnRjpGOi8lLGFjY2VudHVuZGVyR0Y/RistSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuNGVtRicvJSZkZXB0aEdGZG8vJSpsaW5lYnJlYWtHUSVhdXRvRidGKy1GIzYmLUY3Ni1RIihGJ0Y6L0Y+RkpGQC9GQ0ZKRkRGRkZnbkZLRk1GaG4tSSZtZnJhY0dGJDYoLUYjNiZGKy1GIzYnLUYsNiVRJHNpbkYnRlhGOi1GNzYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y6Rj1GQEZCRkRGRkZnbkZLRk1GaG4tRiM2JkZfcC1GIzYkRlVGOi1GNzYtUSIpRidGOkZicEZARmNwRkRGRkZnbkZLRk0vRlFRLDAuMTExMTExMWVtRidGOkYrRjpGK0Y6RmNxLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zfci8lKWJldmVsbGVkR0Y/RmVxRjpGK0Y6LUY3Ni1RIj1GJ0Y6Rj1GQEZCRkRGRkZnbkZLL0ZOUSwwLjI3Nzc3NzhlbUYnL0ZRRmhyLUZqbjYkUSIxRidGOkY6RitGOg==</Equation> </Text-field>Figuras 4.5 e 4.6 ilustram que a \303\241rea ( area ) do tri\303\242ngulo POB \303\251 limitada acima ( bounded above )pela \303\241rea do setor POA e limitada abaixo ( bounded below ) pela \303\241rea do ssetor COB.
Quando o \303\242ngulo feito com o eixo- LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= decresce at\303\251 zero, todas as tr\303\252s \303\241reas tamb\303\251m decrescem at\303\251 zero. Isto \303\251 claramente mostrado pela anima\303\247\303\243o na Figura 4.7.
Figura 4.7 Anima\303\247\303\243o em suporte da prova de que 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
Observe que, para \303\242ngulos pequenos, as tr\303\252s \303\241reas parecem ser iguais.F\303\263rmulas para as tr\303\252s \303\241reas podem ser encontradas da geometria b\303\241sica, e s\303\243o, respectivamente,
que combinadas 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que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUScmIzk1MjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVElJm5lO0YnRjcvJSZmZW5jZUdGNi8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y2LyUqc3ltbWV0cmljR0Y2LyUobGFyZ2VvcEdGNi8lLm1vdmFibGVsaW1pdHNHRjYvJSdhY2NlbnRHRjYvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZOLUkjbW5HRiQ2JFEiMEYnRjdGN0YrRjc= e 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, ent\303\243o \303\251 permitido multiplicar estas inequa\303\247\303\265es ( inequalities ) pela quantidade diferente de zero 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 e 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Para qualquer LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy em (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 ,LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEnJiM5NjA7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y6LUYjNiQtSSNtbkdGJDYkUSIyRidGOkY6LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZILyUpYmV2ZWxsZWRHRjlGK0Y6 ) (excluindo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUScmIzk1MjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiPUYnRjcvJSZmZW5jZUdGNi8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y2LyUqc3ltbWV0cmljR0Y2LyUobGFyZ2VvcEdGNi8lLm1vdmFibGVsaW1pdHNHRjYvJSdhY2NlbnRHRjYvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZOLUkjbW5HRiQ2JFEiMEYnRjdGN0YrRjc= ), 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 > 0 e assim 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 . Isto 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Figura 4.8 confirma a corre\303\247\303\243o -- e utilidade -- destas inequa\303\247\303\265es.
plot([1/cos(x), sin(x)/x, cos(x)], x=-1..1, color=[blue,red,green], legend =["y = 1/cos(x)", "y = sin(x)/x", "y = cos(x)"] );Figura 4.8 Teorema do Confronto para 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 em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIRkpGTUY1LUkjbW5HRiQ2JFEiMEYnRjlGNS1GLDYlUSJjRidGL0YyRjk=
Conseq\303\274entemente, porque 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 = 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 = 1, o Teorema do Confronto \303\251 aplic\303\241vel e nos 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
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Prova de que <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> </Text-field>Esta prova ( proof ) come\303\247a pela multiplica\303\247\303\243o de 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 por 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 e ent\303\243o usando identidades trigonom\303\251tricas para simplificar:
Portanto,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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KEYrLUYjNiYtSSNtb0dGJDYtUSIoRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSV0cnVlRicvJSpzZXBhcmF0b3JHUSZmYWxzZUYnLyUpc3RyZXRjaHlHRjwvJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkwtRiM2KkYrLUknbXVuZGVyR0YkNiUtRjQ2LVEkbGltRidGNy9GO0Y/Rj0vRkFGP0ZCRkQvRkdGPEZIRkovRk5RLDAuMTY2NjY2N2VtRictRiM2Ji1GLDYlUScmIzk1MjtGJy8lJ2l0YWxpY0dGP0Y3LUY0Ni1RJyZyYXJyO0YnRjdGV0Y9RlhGQkZERkZGSEZKRk0tSSNtbkdGJDYkUSIwRidGN0Y3LyUsYWNjZW50dW5kZXJHRj9GKy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC40ZW1GJy8lJmRlcHRoR0ZbcC8lKmxpbmVicmVha0dRJWF1dG9GJ0YrLUYjNiZGMy1JJm1mcmFjR0YkNigtRiM2JkYrLUYjNictRiw2JVEkc2luRidGW29GNy1GNDYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y3RldGPUZYRkJGREZGRkhGSkZNLUYjNiZGMy1GIzYkRmhuRjctRjQ2LVEiKUYnRjdGOkY9RkBGQkZERkZGSEZKL0ZOUSwwLjExMTExMTFlbUYnRjdGK0Y3RitGN0ZlcS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGYXIvJSliZXZlbGxlZEdGP0ZncUY3RitGN0ZncUY3LUY0Ni1RMSZJbnZpc2libGVUaW1lcztGJ0Y3RldGPUZYRkJGREZGRkhGSkZNLUYjNiZGMy1GIzYqRitGUUYrRmZvRistRiM2JkYzLUZncDYoRmlwLUYjNiZGKy1GIzYnLUZhbzYkUSIxRidGNy1GNDYtUSIrRidGN0ZXRj1GWEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORlx0LUYjNictRiw2JVEkY29zRidGW29GN0ZgcUZjcUYrRjdGK0Y3RitGN0ZcckZfckZickZkckZncUY3RitGN0ZncUY3RitGN0YrRjc= = (1)(0) = 0 Quest\303\243o 4.1
Encontre o limite: 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 o limite \303\251 indefinido (ou infinito), digite a sua resposta como dne.Quest\303\243o 4.2
Encontre o limite: 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 o limite \303\251 indefinido (ou infinito), digite a sua resposta como dne.Quest\303\243o 4.3
Encontre o limite: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY0LUkjbWlHRiQ2JVElbGltdEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY1RjVGNUY1LUkmbWZyYWNHRiQ2KC1GIzYmLUYsNiVRJHNpbkYnL0YwRj1GOS1JKG1mZW5jZWRHRiQ2JC1GIzYnLUkjbW5HRiQ2JFEiNUYnRjlGNS1GLDYlUSJ4RidGL0YyLyUrZXhlY3V0YWJsZUdGPUY5RjlGXm9GOS1GIzYlRltvRi9GMi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGZ28vJSliZXZlbGxlZEdGPS1GLDYjUSFGJy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC4wZW1GJy8lJmRlcHRoR0ZkcC8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GLDYlRl1vRldGOS1GNjYtUSgmc3JhcnI7RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS1GaG42JFEiMEYnRjlGNUZfcC1GYHA2JkZicEZlcEZocC9GW3FRJWF1dG9GJ0ZccEZeb0Y5Se o limite \303\251 indefinido (ou infinito), digite a sua resposta como dne.
<Text-field style="Heading 1" layout="Heading 2" spaceabove="7"><Font encoding="UTF-8">Cap\303\255tulo Resumo </Font></Text-field>Limites envolvendo fun\303\247\303\265es trigonom\303\251tricas fundamentais seno e cosseno s\303\243o f\303\241ceis de ser avaliadas conquanto divis\303\243o por zero seja evitada e n\303\243o exista oscila\303\247\303\243o infinita no ponto de limite. (No Cap\303\255tulo 7 na continuidade ( continuity ) vamos aprender que as fun\303\247\303\265es seno e o cosseno s\303\243o cont\303\255nuas para todos os n\303\272meros reais ( real numbers).) Quando divis\303\243o por zero \303\251 encontrada, simplifica\303\247\303\265es via fatoriza\303\247\303\243o ( factorization ) podem ser tentadas. Em muitos casos, entretanto, isto n\303\243o ser\303\241 suficiente para avaliar um limite. Nestas situa\303\247\303\265es, deve-se lembrar dos dois limites 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 e 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 que estas express\303\265es n\303\243o apare\303\247am explicitamente (explicitly ) no limite original, lembre-se de explorar diferentes caminhos de que voc\303\252 j\303\241 deve ser capaz , para conduzir a express\303\243o a uma forma na qual um desses limites possa ser aplicado.Os m\303\251todos usados na avalia\303\247\303\243o rigorosa dos dois limites especiais --- geometria e o Teorema do Confronto --- podem ser usados para outros problemas. Encontrar os limites ( bounds ) superiores e inferiores apropriados para o Teorema do Confrontopode ser um desafio. Um gr\303\241fico pode \303\240s vezes ajudar, indicando escolhas razo\303\241veis para os limites de fun\303\247\303\265es.
<Text-field style="Heading 1" layout="Heading 2" spaceabove="7"><Font encoding="UTF-8">Qual \303\251 o Pr\303\263ximo </Font></Text-field>Estamos no meio da Unit 1. Este \303\251 um bom momento para rever todos os conceitos contidos nos quatro primeiros cap\303\255tulos.A segunda metade da unidade examina limites no infinito, limites infinitos,e continuidade antes de concluir com uma das aplica\303\247\303\265es te\303\263ricas essenciais de continuidade, o teorema do valor m\303\251dio ( intermediate value theorem ).Chapter Navigation<< Prev123 4 5678910111213141516171819202122232425262728293031Next >>
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