Limites trigonometricos

Limites Trigonométricos Resolvidos
Sete páginas e 34 limites resolvidos senx =1 x→0 x

Usar o limite fundamental e alguns artifícios :

lim

0 x x lim = , é uma indeterminação.
=? à x →0 sen x x → 0 sen x
0
x
1
1 x lim
= lim
=
= 1 logo lim
=1
sen x x →0 sen x x →0 sen x x → 0 sen x lim x→0 x x sen 4 x sen 4 x sen 4 x 0 sen y à lim 4.
= 4. lim
= ? à lim
=
=4.1= 4
2. lim x →0 y →0 x→0 x→0
0
4x y x x sen 4 x lim =4 x→0 x sen 5 x
5 sen 5 x
5 sen y 5 sen 5 x
= ? à lim . logo lim
3. lim
= lim .
=
x →0 2 x x →0 2 y →0 2 x →0 2 x y 5x
2
1. lim

sen mx
=
x →0 nx 4. lim

sen 3 x x →0 sen 2 x

5. lim

x→0

logo senmx = sennx ?à

sen y m . lim n y →0 y

=

=

m m .1= n n

=

5
2

sen mx m
=
x →0 nx n sen y sen 3 x sen 3 x sen 3 x lim 3. lim sen 3 x
3 y →0 y
3
x →0 3 x
3x = . lim = lim x = lim
=.
= .1 = sen t sen 2 x 2 x →0 sen 2 x x → 0 sen 2 x x→0 sen 2 x
2
lim lim 2. x→0 2 x t →0 t x 2x sen 3 x 3 lim = x →0 sen 2 x
2
sen mx sen mx sen mx
m.
sen mx x mx = lim m . mx = m lim = lim
= lim
Logo
sen nx x →0 sen nx x →0 n sen nx x → 0 sen nx x →0 n n. nx nx x ? à lim

=? à

3
2

6. lim

sen mx m sen mx
= lim . x →0 x→0 n nx mx

logo

logo lim

senmx m
=
x → 0 sennx n lim

7.

8.

sen x
0
tgx tgx tgx sen x 1 lim =? à lim
=
= lim cos x = lim
.=
à lim x→ 0 x x→ 0 x x→ 0 x x→ 0 x → 0 cos x x
0
x tgx sen x
1
sen x
1
=1 lim .
= lim
. lim
= 1 Logo lim x→ 0 x→ 0 x → 0 cos x x→ 0 x x cos x x x → 1
0
tg (t ) tg a 2 − 1 tg a 2 − 1
= ? à lim 2
=
lim à Fazendo t = a 2 − 1,  à lim
=1
2 a →1 a − 1 a →1 a − 1 t →0 t t →0
0


(

)

logo lim

a →1

(

(

)

) =1

tg a 2 − 1 a2 −1

1

Limites Trigonométricos Resolvidos
Sete páginas e 34 limites resolvidos
9. lim

x →0

x − sen 3 x x + sen 2 x

= ? à