Analisi 1 - Schema funzioni limiti notevoli e McLaurin

FUNZIONI

LIMITI NOTEVOLI

[1] sin x

∘ =  ∽ = + →

lim 1 sin x x; sin x x ox, x 0

x x→0

x→0 tan x

∘ =  ∽ = + →

lim 1 tan x x; tan x x ox, x 0

x x→0

x→0 − 2

1 cos x x

∘ =  − ∽ = − + , →

1 1 2 2

lim 1 cos x x ; cos x 1 ox x 0

2 2 2 2

x x→0

x→0 arctan x

∘ =  ∽ = + →

lim 1 arctan x x; arctan x x ox, x 0

x x→0

x→0 arcsin x

∘ =  ∽ = + →

lim 1 arcsin x x; arcsin x x ox, x 0

x x→0

x→0

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α α

+ αx = α ∈  + αx ∽

∘ 1 1

1/x 1/x

lim e , R e ;

x→0

x→0 α

+ αx = + →

1 1/x e o1, x 0

+

log1 x

∘ =  + ∽ + = + →

lim 1 log1 x x; log1 x x ox, x 0

x x→0

x→0 1 +

log x 1 x

= a > ≠  1 + ∽ 1 ≠ >

∘ a

lim , 0, a 1 log x , a 0;

x a

log a log a

x→0

x→0 x

1 + = + → a > ≠

log x ox, x 0 0, a 1

a log a

−

x

e 1

∘ =  − ∽ = + + →

x x

lim 1 e 1 x; e 1 x ox, x 0

x x→0

x→0 −

x

a 1

∘ = a >  − ∽ 1 ≠ >

x

lim log a, 0 a 1 x log a, a 0;

x x→0

x→0 = + + → 1 ≠ >

x

a 1 x log a ox, x 0 a 0

α

1 + −

x 1 α

∘ = α, α ∈  1 + − ∽ αx α ≠

lim R x 1 0;

x x→0

x→0 α

1 + = + αx + → α ≠

x 1 ox, x 0 0

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sinh x

∘ =  ∽ = + →

lim 1 sinh x x; sinh x x ox, x 0

x x→0

x→0 tanh x

∘ =  ∽ = + →

lim 1 tanh x x; tanh x x ox, x 0

x x→0

x→0 − 2

cosh x 1 x

∘ =  − ∽ = + + , →

1 1 2 2

lim cosh x 1 x ; cosh x 1 ox x 0

2 2 2 2

x x→0

x→0

-----------------------------------------------------------

∘ = a > ∀b ∈ 

b

a

lim x x| 0 0, R

|log

+

x→0 −a +

= , → a > ∀b ∈

b

x| ox x 0 , 0, R

|log b

∘ = γ ∈ 0, ∪ 1, +∞; > ∀b ∈ 

a

lim x log x 0 1 a 0, R

γ

+

x→0 b −a +

= , → γ ∈ 0, ∪ 1, +∞; > ∀b ∈

log x ox x 0 , 1 a 0, R

γ b

∘ = γ ∈ 0, ∪ 1, +∞, > ∀b ∈ 

a

lim log 0 1 a 0, R

|x| |x|

γ

x→0 b −a

= , → γ ∈ 0, ∪ 1, +∞; > ∀b ∈

log o|x| x 0, 1 a 0, R

|x|

γ

——————————————————————————————————–

[2] ax ax

x x

α α

∘ + = α ∈  + ∽

lim 1 e , R 1 e ;

x→+∞

x→+∞ ax x α

+ = +