1
Matemática 12ª C1 e 2
Limites notáveis
Limite trigonométrico
Consideremos a função f continua, de domínio IR\ {0}, definida por:
f ( x) 
O cálculo de lim
x 0
sen x
x
sen x
0
conduz-nos à indeterminação
x
0
lim
x 0
Generalizando: lim
x 0
sen x
1
x
sen k x
1
kx
Exemplos
a) lim
sen 4 x
sen 4 x
sen 4 x
 lim 4.
 4. lim
 4.1=4
x 0
x 0
x
4x
4x
b) lim
sen 5 x
5 sen 5 x 5
sen 5 x 5
5
 lim
 lim
 .1 
x 0
2x
2 5x
2 x 0 5 x
2
2
x 0
x 0
sen 2 x
tg 2 x
sen 2 x
sen 2 x 1
cos 2 x
 lim
lim
 lim
.

c) lim

x 0
x 0
x 0
x cos 2 x x0 x
cos 2 x
x
x
lim 2
x 0
1
sen 2 x 1
sen 2 x 1
.
 2 lim
.
 2.1.  2
x 0
2 x cos 2 x
2 x cos 2 x
1
2
1  cos x 0
 (vamos multiplicar e dividir pelo conjugado)
x 0
x
0
(1  cos x)(1  cos x)
1  cos2 x
sen2 x
lim
 lim

lim

x 0
x 0
x(1  cos x )
x(1  cos x) x0 x(1  cos x)
sen x
sen x
0
lim
.lim
 1.
0
x 0
x x 0 1  cos x
11
d) lim
Exercícios
Resolva os seguintes limites:
sen2 x
a) lim
x 0
x
sen 2 x  sen x
b) lim
x 0
2x
sen x  sen 5 x
c) lim
x 0
sen 2 x  sen 4 x
d) lim
cos x  1
3x 2
e) lim
sen x . sen 3 x . sen 5 x
x3
x 0
x 0
Limite de Nepper
x
 1
lim 1    e
x  
x
x
Generalizando:
 k
lim 1    e k
x 
 x
3
Exemplos:
 2
  2
a ) lim 1    lim 1 
 e 2


x  
x
x 

x
x
1
2
1
3

3
3 

3 2
b) lim 1    lim1     (e )  e 2
x 
 2 x  x  2 x  
x
2x
1
 x  3
 3
3
c) lim 
  lim 1    e  3
x  
x 
e
 x
n
x
x
 1
O lim 1   tem uma indeterminação  1
x  
x
Sejam f ( x) e g ( x) duas funcoes, tais que lim f ( x)  1e lim g ( x )   então:
x a
x a
lim  f ( x)
g ( x)
x a
Exemplos:
a) lim 1  x   1 
2
x
x 0
e
lim
x0
1  x   1 2
x

b) lim 1 
x  
e
e
 e 2 
1
e2
x
4
2

 1
x
2  x

lim   1    1 .
x  4

x 0
 2x 
lim   
x 
x 0 
e
 2x 
lim  
x 0  4 x 
1
2
e  e
 e
lim  f ( x ) 1 g ( x )
x a
4
Exercícios
Calcule os limites:
x
 x  1
a ) lim 

x  
 x 
 3
c) lim 1  
x  
x

 5x  2 
e) lim 

x  
 5x  1 
 x 1 
g ) lim 

x  
 x  3
 x 1
i ) lim 

x  
 x  1
x
4x

1 

b) lim 1 
x  
2 x 

 x 
d ) lim 

x  
 x  3
3x
 x2 1 
f ) lim  2

x  
 x  2
x2
 x  5
h) lim 

x  
 x 1
3 x2
2x
 x3 
j ) lim 

x  
 2x  1
x2
x
2 x 5
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Ficha de exercícios
1. Calcula os seguintes limites:
xk
 1
1.1. lim 1  
x 
 x
 x 1
1.4. lim 

x 
 x 
 1
1.2. lim 1  
x 
 x
xk
x
1 

1.5. lim 1  
x 
 2x 
x
 x 1 
1.8. lim 

x 
 x  2
 x 
1.7. lim 

x 
 x  3
 x 1
1.10. lim 

x 
 x 1
2 x 5
3x2
2x
 3
1.16. lim 1  
x 
x

x
 x2  x  3 

1.17. lim  2
x 
 x 2 
 x3  1 

1.19. lim  3
x 
 x 2
x3
 x  3
1.20. lim 

x 
 x 1 
2x
 x 1 
1.12. lim 

x 
 x  3
x2
 3x  1 
1.15. lim 

x 
 x 1 
x
x 3
x
4x
 x 5
1.9. lim 

x 
 x 1 
3 x
1.13. lim 

x 
 x 
 x 
1.14. lim 

x 
 x 1
 3
1.6. lim 1  
x 
 x
x
 x2 1 

1.11. lim  2
x 
 x 2
x
 k
1.3. lim 1  
x 
 x
x 5
x 1
 x  10 
1.18. lim 

x 
 x  10 
x
 2x  1 
1.21. lim 

x 
 2x  3 
2x
2. Calcula os seguintes limites trigonométricos:
2.1. lim
sen 5 x
x
2.2. lim
sen ( x )
x
2.3. lim
sen ( x)
2x
2.4. lim
x  sen x
x  sen
2.5. lim
sen a x
sen b x
2.6. lim
sen x  x
x
2.7. lim
1  cos x
x2
2.8. lim
sen2 x  x 2
x( senx  x)
2.9. lim
x 0
x 0
x 0
2.10. lim
x

4
x 0
x 0
x 0
sen x  cos x
1  tg x
 x 2  2x  3 

2.13. lim  2
x 0
 x  3x  2 
2.11. lim
x 1
sen x
x
2.14. lim
x 0
sen( x )
sen (3 x)
1  cos x
x
x 0
x 0
1  cos (2 x)
3 
sen x 
2 
x sen x
2.12. lim
x  0 1  cos x
x 0
2.15. lim
x 0
x 2  sen2 (3x)
x2