Limites Envolvendo a Seqüência de Euler Leohnard Euler

Prévia do material em texto

�PAGE �
		
Limites Envolvendo a Seqüência de Euler 
Leohnard Euler, 1707-1783, ( 27/08/01 11:50:50, arquivo: aueulog.doc 
�
Usar o limite fundamental: 
 ou 
Limites da Seqüência de Euler:	
�� EMBED Equation.3 = ? 	( 
=
=
= 
= 
= 
 Fazendo t= 
 e x=2t
= ? 	(	
= 
 = 
= 
 = 
= 
Fazendo 
e 
=? 		( 
= 
= 
= 
.
=
 
com 
=? 		( 
=
= 
= 
.
= 
=
= 
= 
=
			Fazendo t=
 e x=
.
=?			( 
=
= 
= 
= 
= 
=?			( 
= 
= 
= 
= 
=
=e2a .
=?			(	
= 
=?			(	
= 
=
= 
= 
=
 
fazendo 
 , com t=-3y 
=?			( fazendo 
 e substituindo no limite, temos 
=
=
Outros Limites	que dependem do limite fundamental 
�� EMBED Equation.3 
 �
Teorema: Prove que 
�� EMBED Equation.3 , onde 
 é o logaritmo de a na base e.�
Demonstração: Fazendo 
, 
 
, aplicando-se logaritmo natural na igualdade anterior, temos: 
(
( 
	Logo 
= 
= 
= 
= 
= 
= 
= 
.
�
Calcular os limites abaixo: use o limite 	
�� EMBED Equation.3 
=?			(	
= 
=
=
.
Fazendo 
 com x=
.
=?			( 
= 
= 
= 
= 
= 
=
=
=
= ?			(	
=
= 
= 
= 
= ?			(	
= 
= 
= 
.lne= 
=? 			(	
= 
= 
=
= 
=
=
 		Fazendo 
, onde 
.
=?			(	
= 
= 
= 
=?			(	
= 
= 
= 
= 
=?			(	
= 
= 
= 
= 
=?		(	Solução: 
= 
= 
= 
= 
= 
=
= 
= 
BriotxRuffini para fatorar: 
=?		= 
2
5
2
-2
•
-4
-2
2
1
0 (resto)
=?			(	
= 
=
= 
=
=
=?			(	
= 
=
=
 =
= 
=?			(	
= 
= 
= 
=
=?			(	
= 
= 
= 
=
=?			(	
=
=
=
=?			(	
= 
=
 =
= 
= 
 = 
= 
=?		(	
= 
= 
= 
= 
= 
=
= 
 		Fazendo 
=?			(	
= 
= 
= 
 = 
= 
= 2.
Aplicando a Regra de L’Hôspital: 
= ?		( 
= 
Fazendo 
	( 
= 
. Derivando separadamente o numerador e o denominador, temos:	
 e 
= 
.	Logo 
= 
=2.
=?			(	
= 
= 
 = 
 =
= 
= 
= 
=
= 0
�PAGE �
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