Limites-notaveis 5s49t

Limites Notáveis!!! Indeterminações ∞ 0 ∞−∞, , , 0× ∞ ∞ 0 Recorda, para a > 1 , lim ( a x ) = 0

lim ( a x ) = +∞

x →−∞

x

lim

x →+∞

lim ( log a x ) = −∞ log a ( x )

p

a = +∞ ,p ∈ ℝ xp

lim

x →+∞

lim ( log a x ) = +∞

x → 0+

x →+∞

x = 0 ,p ∈ ℝ ax

lim

ebx − 1 = 1, b ≠ 0 x →0 bx

lim

xp = +∞ ,p > 0 x →+∞ log ( x ) a lim

= 0 ,p > 0

xp

x →+∞

x →+∞

lim

ln ( bx + 1)

x →0

bx

=1 , b ≠ 0

Limites Notáveis: ex = +∞ ,p ∈ ℝ x →+∞ x p lim

lim

ln ( x )

lim

ln ( x + 1)

x →+∞

ex −1 =1 x →0 x

lim

x →0

x x

=0 =1

Usando os limites indicados, calcula os seguintes limites: e h + 3 − e3 2. lim h →0 h

e x − e2 x 1. lim x →0 5x 2x + x2 x →0 x + 1 ln ( x + 2 ) − ln 2

x →0

10. lim

2x log ( x + 2 ) − log 2

x →0

2x

13. lim

ln ( 3 x )

17. lim

ln ( 3 + x )

x →+∞

x →+∞

Calculo de limite

1

1 x →−∞ xe x

5. lim

4. lim 7. lim

 1  3. lim ( 2n + 1)  e n − 1 n →+∞  

x

x +1

8. lim

6. lim+ x 2 e x x →0

ln ( x + 1)

x →0

11. lim x →0

3

2x ln (1 + 3 x ) 4x

9. lim x →3

x2 − 5x + 6 ln ( x − 2 )

12. lim

x →−5

ln ( 6 + x ) x+5

14. lim

e2 x − 1 x → 0 ln ( 3 x + 1)

 1 15. lim 2 x.ln 1 +  x →+∞  x

18. lim ln ( x + 1) − x  x →+∞

19. lim f ( x ) , sendo x →1

 2 x2 − x  4 x − 1 , x ≤ 1 f ( x) =   ln ( 2 x − 1) , x > 1  x − 1

Prof. Eva Figueiredo