Limites

01.

•Limites Fundamentais

Alguns limites fundamentais senx lim
1
x 0 x x

senx

senx/x

0,5

x 0

0,4

x

1

lim  1    e x   x 

0,389418342

0,973545855

0,3

senx
1
x

0,958851077

0,295520206

0,985067355

0,2

0,198669330

0,993346654

0,1

0,099833416

0,998334166

0,01

0,009999833

0,999983333

0,001 0,000999999

lim

0,479425538

0,999999833

Adalberto Santos

Aplicações:
a)

b)

lim x 0

lim x 0

sen2 x
?
x

Aplicações:
 lim 2 x 0

sen2 x
2x

 2.1  2

lim

x 0

1  cos x
?
x2

 lim

x 0

( 1  cos x )( 1  cos x ) x 2 ( 1  cos x )

3 sen3x
5x
sen3x
 ? lim .
.
 x 0 5 sen5x 3x sen5x 3
3
.1.1 
5
5

1

Mais limites fundamentais

Aplicações:

x

02.

1

lim  1    e x   x  x 10
100
1000
10000
100000
1000000
......
......
109

2



1+ 1/x
(1+1/x)x
1,1
2,593
1,01
2,704813
1,001
2,716923
1,0001
2,718145
1,00001
2,718268
1,000001
2,718280
.....
......
......
......
1,000000001 2,71828182

1

2x

a) xlim 1  x 




x

1  lim 1    
 ? x  x 




e2

x

3

lim
b) x 1    ? fazendo x  3w x 
3 

lim 1 

w 
3w 


3w

3

w

1 
 lim 1     w 
 w  



e3

x

 x 

1

lim
d) x 1  x   ? fazendo x  w

 w 1

w
1 
1
1

lim lim 1    w  1  w    e
  w 



 w

x

lim
c) x  x  1   ?


 x  1 lim 

x 
 x 

x

 x 1  lim    x  
 x x 


x

 x  1  x 
 lim 
  x  
 x  







1

1



x


1  lim  x   1    x 



x

e)

1



e 1

 x  1 lim 
 ? x  x  1



1

lim 1   x   x  x 1

lim 1   x   x 