1. Limites Trigonométricos Resolvidos
Sete páginas e 34 limites resolvidos
senx
Usar o limite fundamental e alguns artifícios : lim =1
x→0 x
x x 0
1. lim =? à lim = , é uma indeterminação.
x → 0 sen x x →0 sen x 0
x 1 1 x
lim = lim = = 1 logo lim =1
x →0 sen x x →0 sen x sen x x → 0 sen x
lim
x x→0 x
sen 4 x sen 4 x 0 sen 4 x sen y
2. lim = ? à lim = à lim 4. = 4. lim =4.1= 4 logo
x→0 x x→0 x 0 x →0 4x y →0 y
sen 4 x
lim =4
x→0 x
sen 5 x 5 sen 5 x 5 sen y 5 sen 5 x 5
3. lim = ? à lim . = lim . = logo lim =
x →0 2 x x →0 2 5x y →0 2 y 2 x →0 2 x 2
sen mx sen mx m sen mx m sen y m m sen mx m
4. lim = ? à lim = lim . = . lim = .1= logo lim =
x →0 nx x →0 nx x→0 n mx n y →0 y n n x →0 nx n
sen 3 x sen 3 x sen 3 x sen y
3. lim lim
sen 3 x sen 3 x x →0 3 x 3 y →0 y 3
5. lim =? à lim = lim x = lim 3x = . = . = .1 =
x →0 sen 2 x x →0 sen 2 x x → 0 sen 2 x x→0 sen 2 x sen 2 x 2 sen t 2
2. lim lim
x 2x x→0 2 x t →0 t
3 sen 3 x 3
logo lim =
2 x →0 sen 2 x 2
sen mx sen mx sen mx
m.
senmx sen mx x mx = lim m . mx = m
6. lim = ? à lim = lim = lim Logo
x→0 sennx x →0 sen nx x → 0 sen nx x →0 sen nx x →0 n sen nx n
n.
x nx nx
senmx m
lim =
x → 0 sennx n
sen x
tgx tgx 0 tgx sen x 1
7. lim = ? à lim = à lim = lim cos x = lim . =
x→ 0 x x→ 0 x 0 x→ 0 x x→ 0 x x → 0 cos x x
sen x 1 sen x 1 tgx
lim . = lim . lim = 1 Logo lim =1
x→ 0 x cos x x→ 0 x x → 0 cos x x→ 0 x
8. lim
(
tg a 2 − 1 )
= ? à lim 2
tg a 2 − 1 (=
0 ) x → 1
à Fazendo t = a 2 − 1,  à lim
tg (t )
=1
a →1 a − 1
2 a →1 a − 1 0 t →0 t →0 t
logo lim
( ) =1
tg a 2 − 1
a →1 a2 −1
1

2. Limites Trigonométricos Resolvidos
Sete páginas e 34 limites resolvidos
 sen 3 x 
x.1 − 
x − sen 3 x x − sen 3 x x − sen 3 x  x 
à f (x ) =
0
9. lim = ? à lim = = =
x →0 x + sen 2 x x →0 x + sen 2 x 0 x + sen 2 x  sen 5 x 
x.1 + 
 x 
 sen 3 x  sen 3 x sen 3 x
x.1 − 3.  1 − 3. 1 − 3.
 3. x 
= 3. x à lim 3.x = 1 − 3 = −2 = − 1 logo
 sen 5 x  sen 5 x x →0 sen 5 x 1+ 5 6 3
x.1 + 5.  1 + 5. 1 + 5.
 5. x  5. x 5. x
x − sen 3 x 1
lim =−
x →0 x + sen 2 x 3
tgx − sen x tgx − sen x sen x 1 sen 2 x 1 1
10. lim = ? à lim = lim . . . =
x →0 x 3 x →0 x 3 x→0 x cos x x 2 1 + cos x 2
sen x sen x − sen x. cos x
− sen x
tgx − sen x cos x sen x.(1 − cos x ) sen x 1 1 − cos x
f (x ) = = = cos x = = . . =
x 3
x3 x3 x 3 . cos x x x 2 cos x
sen x 1 1 − cos x 1 + cos x sen x 1 1 − cos 2 x 1 sen x 1 sen 2 x 1
. . . = . . . = . . .
x x 2 cos x 1 + cos x x cos x x 2 1 + cos x x cos x x 2 1 + cos x
tgx − sen x 1
Logo lim =
x →0 x3 2
1 + tgx − 1 + sen x tgx − sen x 1
11. lim =? à lim . =
x →0 x 3 x →0 x 3
1 + tgx + 1 + sen x
sen x 1 sen 2 x 1 1 1 1 1 1 1
lim . . . . = 1. . . . =
x →0 x cos x x 2 1 + cos x 1 + tgx + 1 + sen x 1 1 2 2 4
1 + tgx − 1 + senx 1 + tgx − 1 − sen x tgx − sen x
f (x ) =
1 1
= . = .
x3 x3 1 + tgx + 1 + sen x x3 1 + tgx + 1 + sen x
1 + tgx − 1 + sen x 1
lim 3
=
x →0 x 4
x−a x+a
2 sen . cos 
sen x − sen a sen x − sen a  2   2 =
12. lim =? à lim = lim
x→a x−a x→a x−a x→a x−a
2. 
 2 
x − a . cos x + a 
 
2 sen( )
2 .  2  sen x − sen a
lim = cos a Logo lim = cosa
x→a x−a 1 x→a x−a
2. 
 2 
2

3. Limites Trigonométricos Resolvidos
Sete páginas e 34 limites resolvidos
 x+a−x  x+a+ x
2 sen  . cos 
sen ( x + a ) − sen x sen ( x + a ) − sen x  2   2 
13. lim = ? à lim = lim . =
a →0 a a →0 a a→a  x−a 1
2. 
 2 
a  2x + a 
2 sen  . cos 
 2  2  sen ( x + a ) − sen x
lim . = cos x Logo lim =cosx
a→a a 1 a →0 a
2. 
2
x+a+ x x−a− x
− 2 sen . sen 
cos( x + a ) − cos x cos( x + a ) − cos x  2   2 
14. lim = ? à lim = lim =
a→0 a a→0 a a →0 a
 2x + a  −a −a
− 2. sen . sen  sen 
2x + a 
lim
 2   2 
= lim − sen  .
 2 
= − sen x Logo
a→0 −a a→0  2  −a
2.   
 2   2 
cos( x + a ) − cos x
lim =-senx
a→0 a
1 1 cos a − cos x
−
sec x − sec a sec x − sec a cos x cos a
15. lim = ? à lim = lim = lim cos x. cos a =
x→a x−a x→a x−a x→ a x−a x→a x−a
a+ x a−x
− 2. sen . sen  
cos a − cos x  2   2 =
lim = lim
x → a ( x − a ). cos x. cos a x→a (x − a ). cos x. cos a
a+ x a−x a+x a−x
− 2. sen   sen  sen  sen 
 2 .  2 . 1  2   2  1
lim = lim . . =
x→a 1  a − x  cos x. cos a x→ a 1  a − x  cos x. cos a
− 2.   
 2   2 
sen a 1 sen a 1 sec x − sec a
.1 . = . = tga. sec a Logo lim = tga. sec a
1 cos a. cos a cos a cos a x→a x−a
16. lim
x2
x → 0 1 − sec x
= ? à lim
x2
x → 0 1 − sec x
= lim
x→0 2
1
= −
2
sen x 1 1
− . .
x2 cos x (1 + cos x )
x2 x2 x 2 . cos x
f (x ) =
1
= = = =
1−
1 cos x − 1 − 1.(1 − cos x )
−
(1 − cos x ) . 1 . (1 + cos x )
cos x cos x x2 cos x (1 + cos x )
1 1
=
1 − cos 2 x 1 1 sen 2 x 1 1
− . . − . .
x 2 cos x (1 + cos x ) x 2 cos x (1 + cos x )
3

4. Limites Trigonométricos Resolvidos
Sete páginas e 34 limites resolvidos
1 tgx − 1
1−
1 − cot gx 1 − cot gx tgx tgx
17. lim =?à lim = lim = lim =
π 1 − tgx π 1 − tgx
x → 1 − tgx
π π 1 − tgx
x→ x→ x→
4 4 4 4
−1.(1 − tgx )
tgx 1 1 − cot gx
lim = lim − = −1 Logo lim = -1
x→
π 1 − tgx π
x→ tgx π
x→ 1 − tgx
4 4 4
18. lim
1 − cos x
3
= ? à lim
1 − cos x 3
= lim
(1 − cos x ).(1 + cos x + cos 2 x ) =
x→0 sen 2 x x→0 sen 2 x x →0 1 − cos 2 x
lim
(1 − cos x ).(1 + cos x + cos 2 x ) = lim 1 + cos x + cos 2 x = 3 Logo lim
1 − cos 3 x
=
3
x →0 (1 − cos x )(1 + cos x )
. x →0 1 + cos x 2 x → 0 sen 2 x 2
sen 3 x sen 3 x sen x.(1 + 2. cos x )
19. lim = ? à lim = lim − =− 3
π
x→ 1 − 2. cos x π
x→ 1 − 2. cos x
π
x→ 1
3 3 3
f (x ) =
sen 3 x
=
sen ( x + 2 x )
=
sen x. cos 2 x + sen 2 x. cos x
=
(
sen x. 2 cos 2 x − 1 + 2. sen x. cos x. cos x
=
)
1 − 2. cos x 1 − 2. cos x 1 − 2. cos x 1 − 2. cos x
[( )
sen x. 2 cos 2 x − 1 + 2 cos 2 x
=
]
sen x. 4 cos 2 x − 1
=−
[ ]
sen x.(1 − 2.cox )(1 + 2.cox )
.
= −
sen x.(1 + 2. cos x )
1 − 2. cos x 1 − 2. cos x 1 − 2. cos x 1
sen x − cos x sen x − cos x
= lim (− cos x ) = −
2
20. lim =? à lim
x →π
4
1 − tgx x →π
4
1 − tgx x →π
4
2
sen x − cos x sen x − cos x sen x − cos x sen x − cos x sen x − cos x
f (x ) = = = = = =
1 − tgx sen x sen x cos x − sen x − 1.(sen x − cos x )
1− 1−
cos x cos x cos x cos x
sen x − cos x cos x
− . = − cos x
1 cos x − sen x
21. lim (3 − x ). cos sec(πx ) = ? à lim (3 − x ). cos sec(πx ) = 0.∞
x→3 x→3
3− x 3− x
f (x ) = (3 − x ). cos sec(πx) = (3 − x ).
1 1
= = = =
sen (πx ) sen (π − πx ) sen (3π − πx ) π . sen (3π − πx )
π .(3 − x )
1
à lim (3 − x ). cos sec(πx ) = lim
1 1
=
π . sen (3π − πx ) x→3 x→3 π . sen (3π − πx ) π
(3π − πx ) (3π − πx )
1 1
22. lim x. sen( ) = ? à lim x. sen( ) = ∞.0
x→∝ x x→∝ x
1
sen 
x sen t 1  x → +∞
lim = lim =1 à Fazendo t = 
x →∝ 1 t →0 t x t → 0
x
4

5. Limites Trigonométricos Resolvidos
Sete páginas e 34 limites resolvidos
π
2. sen 2 x + sen x − 1 1 + sen
2. sen 2 x + sen x − 1 1 + sen x 6 =
23. lim = ? à lim = lim =
x →π 2. sen x − 3. sen x + 1
2 x →π 2. sen 2 x − 3. sen x + 1 x →π − 1 + sen x π
6 6 6 − 1 + sen
6
 1
1  sen x − .(sen x + 1)
1+
2 =−3 à f (x ) =
2. sen x + sen x − 1
2
=
2
=
(sen x + 1) = 1 + sen x
−1+
1 2. sen x − 3. sen x + 1 
2 1
 sen x − .(sen x − 1)
(sen x − 1) − 1 + sen x
2  2
πx πx πx
24. lim(1 − x ).tg   = ? à lim(1 − x ).tg   = 0.∞ à f (x ) = (1 − x ).tg   =
     
x →1  2 x →1  2  2 
π
(1 − x ) = 2 .(1 − x ). π =
2 2 2
 π πx  π π
(1 − x ). cot g  −  = = à
2 2   π πx   π πx   π πx   π πx 
tg  −  tg  −  tg  −  tg  − 
2 2  2 2  2 2  2 2 
π  π πx 
.(1 − x )  − 
2 2 2 
2 2
 πx  π 2
lim(1 − x ).tg   = lim = π = Fazendo uma mudança de variável,
x →1  2 x →1  π πx  tg (t ) π
tg  −  lim
 2 2  t →0 t
 π πx 
 − 
2 2 
π πx  x → 1
temos : t= − 
2 x t → 0
1− x 2 1− x 2 1+ x 2
25. lim = ? à lim = lim =
x →1 sen (πx ) x →1 sen (πx ) x →1 π . sen (π − πx ) π
(π − πx )
f (x ) =
1− x 2
=
(1 − x )(1 + x ) =
. 1+ x
=
1+ x
=
1+ x
sen πx sen (π − πx ) sen (π − πx ) π . sen (π − πx ) π . sen (π − πx )
(1 − x ) π .(1 − x ) (π − πx )
π  π 
26. lim cot g 2 x. cot g  − x  = ? à lim cot g 2 x. cot g  − x  = ∞.0
x →0
2  x →0
2 
π  tgx 1 − tg 2 x 1 − tg 2 x
f (x ) = cot g 2 x. cot g  − x  = cot g 2 x.tgx =
tgx
= = tgx. =
2  tg 2 x 2tgx 2.tgx 2
1 − tg 2 x
π  1 − tg 2 x 1
lim cot g 2 x. cot g  − x  = lim =
x →0 2  x→0 2 2
cos x − 3 cos x −t2 1
27. lim = lim =−
x →0 sen 2 x t →1 1 + t + t 2 + ... + t 10 + t 11 12
cos x − 3 cos x t3 − t2 − t 2 .(1 − t ) −t2
f (x ) =
(1 − t ).(1 + t + t 2 + ... + t 10 + t 11 )
= = =
sen 2 x 1 − t 12 1 + t + t 2 + ... + t 10 + t 11
x → 0
t = 2.3 cos x = 6 cos x  t 6 = cos x , t 12 = cos 2 x , sen 2 x = 1 − t 12
t → 1
5

6. Limites Trigonométricos Resolvidos
Sete páginas e 34 limites resolvidos
BriotxRuffini :
1 0 0 ... 0 -1
1 • 1 1 ... 1 1
1 1 1 ... 1 0
sen 2 x − cos 2 x − 1 sen 2 x − cos 2 x − 1 π
= lim (− 2. cos x ) = − 2. cos
2
28. lim = ? à lim = − 2. =
π
x→
4
cos x − sen x π x→
4
cos x − sen x π
x→
4
4 2
− 2
f (x ) = =
( =
)
sen 2 x − cos 2 x − 1 2. sen x cos x − 2 cos 2 x − 1 − 1 2. sen x. cos x − 2 cos 2 x + 1 − 1
=
cos x − sen x cos x − sen x cos x − sen x
2. sen x. cos x − 2 cos 2 x − 2. cos x.(cos x − sen x )
= = −2. cos x
cos x − sen x cos x − sen x
sen ( x − 1) sen (x − 1) 1 sen (x − 1) 2 x − 1 + 1
29. lim = ? à lim = lim . . =1
x →1
2x − 1 − 1 x →1 2 x − 1 − 1 x →1 2 (x − 1) 1
sen (x − 1) sen (x − 1) 2x − 1 + 1 sen ( x − 1) 2 x − 1 + 1 sen ( x − 1) 2 x − 1 + 1
f (x ) = = . = . = . =
2x −1 −1 2x − 1 − 1 2x − 1 + 1 2x − 1 − 1 1 2.(x − 1) 1
1 sen (x − 1) 2 x − 1 + 1
. .
2 (x − 1) 1
π − x 
 
sen  3
π + x   2  
1 − 2. cos x 1 − 2. cos x    
30. lim = ? à lim = lim 2. sen 3 . =
π
x→ x−
π π
x→ x−
π x →π  2  π − x 

3 3
3
   3 
3 3  2 
 
 
π +π   2π 

2. sen 3 3 . = 2. sen 3 . = 2. sen π . = 2. 3 = 3
 
 2 
  2 
  3 2
   
π + x  π − x 
   
 1   π  2.(− 2 ) sen 3 . sen 3
2 . − cos x  2 . cos − cos x   2    2  
1 − 2. cos x  =   =    
f (x ) = =  2
3
=
π π π π − x 
x− x − x −  
3 3 3 − 1.2. 3
 2  
 
π + x  π − x  π − x 
     
2. sen 3 . sen 3 sen 3
 2    2   π + x   2  
   
= 2. sen 3 . 
 2  π

π − x    
 3     3−x
 2 
   2 
 
   
1 − cos 2 x 1 − cos 2 x 2. sen x
31. lim =? à lim = limπ
=2
x →0 x. sen x x →0 x. sen x x
x→
3
6

7. Limites Trigonométricos Resolvidos
Sete páginas e 34 limites resolvidos
1 − cos 2 x 1 − ( − 2 sen x ) 1 − 1 + 2 sen 2 x 2. sen x 2. sen x
2 2
1
f (x ) = = = = =
x. sen x x. sen x x. sen x x. sen x x
x x 1 + sen x + 1 − sen x 1+1
32. lim = ? à lim = lim =
x →0 1 + sen x − 1 − sen x x →0 1 + sen x − 1 − sen x x →0 2. sen x 2.1
x
=1
f (x ) =
x
=
(
x. 1 + sen x + 1 − sen x ) = x.( 1 + sen x + 1 − sen x )=
1 + sen x − 1 − sen x 1 + sen x − (1 − sen x ) 1 + sen x − 1 + sen x
(
x. 1 + sen x + 1 − sen x )= 1 + sen x + 1 − sen x
=
1+1
=1
2. sen x sen x 2.1
2.
x
cos 2 x cos x + sen x 2 2
33. lim = lim = + = 2
x→0 cos x − sen x x →0 1 2 2
cos 2 x.(cos x + sen x ) cos 2 x.(cos x + sen x ) cos 2 x.(cos x + sen x )
f (x ) =
cos 2 x
= = = =
cos x − sen x (cos x − sen x )(cos x + sen x )
. cos 2 x − sen 2 x cos 2 x
cos 2 x.(cos x + sen x ) cos x + sen x 2 2
= = + = 2
cos 2 x 1 2 2
 3  π
2. − sen x   
 2  2. sen − sen x 
3 − 2. sen x 3 − 2. sen x
lim   lim  
3
34. lim =? à lim = = =
π π π π π π π π
x→
3 x− x→
3 x− x→
3 x− x→
3 x−
3 3 3 3
 π  π    π − 3x   π + 3x 
  −x  + x      
2. sen 3 . cos 3  2. sen 3 . cos 3 
  2   2    2   2 
         
        
lim = lim  =
π π π 3x − π
x→
3 x− x→
3
3 3
  π − 3x   π + 3x  
2. sen
 . cos 
  6   6 
lim
x→
π − 1.(π − 3 x )
3
3
35. ?
7