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 0x x→0x→0 tan x∘ =  ∽ = + →lim 1 tan x x; tan x x ox, x 0x x→0x→0 − 21 cos x x∘ =  − ∽ = − + , →1 1 2 2lim 1 cos x x ; cos x 1 ox x 02 2 2 2x x→0x→0 arctan x∘ =  ∽ = + →lim 1 arctan x x; arctan x x ox, x 0x x→0x→0 arcsin x∘ =  ∽ = + →lim 1 arcsin x x; arcsin x x ox, x 0x x→0x→0

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α α+ αx = α ∈  + αx ∽∘ 1 11/x 1/xlim e , R e ;x→0x→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 0x x→0x→0 1 +log x 1 x= a > ≠  1 + ∽ 1 ≠ >∘ alim , 0, a 1 log x , a 0;x alog a log ax→0x→0 x1 + = + → a > ≠log x ox, x 0 0, a 1a log a−xe 1∘ =  − ∽ = + + →x xlim 1 e 1 x; e 1 x ox, x 0x x→0x→0 −xa 1∘ = a >  − ∽ 1 ≠ >xlim log a, 0 a 1 x log a, a 0;x x→0x→0 = + + → 1 ≠ >xa 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→0x→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 0x x→0x→0 tanh x∘ =  ∽ = + →lim 1 tanh x x; tanh x x ox, x 0x x→0x→0 − 2cosh x 1 x∘ =  − ∽ = + + , →1 1 2 2lim cosh x 1 x ; cosh x 1 ox x 02 2 2 2x x→0x→0

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∘ = a > ∀b ∈ balim x x| 0 0, R|log+x→0 −a += , → a > ∀b ∈bx| ox x 0 , 0, R|log b∘ = γ ∈ 0, ∪ 1, +∞; > ∀b ∈ alim 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 ∈ alim log 0 1 a 0, R|x| |x|γx→0 b −a= , → γ ∈ 0, ∪ 1, +∞; > ∀b ∈log o|x| x 0, 1 a 0, R|x|γ

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[2]

ax axx xα α∘ + = α ∈  + ∽lim 1 e , R 1 e ;x→+∞x→+∞ ax x α+ = +