Analisi 1 - Schema funzioni limiti notevoli e McLaurin

LIMITI NOTEVOLI

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

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, a ≥ 1

a log a−xe 1∘ = ∅ − ∼ = + + →x → ∞ e 1/x → 0

−xa 1∘ = a > ∅ − ∼ a 1/x o⁢x⁢, x ≥ 0

α∘1 + −x⁢ = α∘, α ∈ ∅ α∘1 + − ∼ αx ∈ R∘ x∘ 1 ≥ 0

α∘1 + = + αx + → α ≠ x⁢, x ≥ 0

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

sinh x∘ = ∅ ∼ = + →lim 1 sinh x/x; sinh x/x o⁢x⁢, x ≥ 0

tanh x∘ = ∅ ∼ = + →lim 1 tanh x/x; tanh x/x o⁢x⁢, x ≥ 0

− 2cosh x/1 x∘ = ∅ − ∼ = + + ∅, →1 1 2 2lim cosh x/1 x ; cosh x/1 o⁢x x 0

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

∀a > ∀b

&x→+∞x→+∞ ax x α+ = + → +∞1 e o1, xax axx xα α∘ + = α ∈  + ∽lim 1 e , R 1 e ;x→−∞x→−∞ ax x α+ = + → −∞1 e o1, x----------------- ----------------------------------------axe∘ = +∞, a, >  = , → +∞ a, >b axlim b 0 x oe x b 0bxx→+∞ β ax = +∞, β > >  = , → +∞ β > >∘ b axlim 1; a, b 0 x oβ x 1; a, b 0bxx→+∞----------------------------------------------------------log ax∘ = a, >  log = , → +∞ a, >a blim 0, b 0 x ox x b 0bxx→+∞ alog xγ = γ ∈ 0, ∪ 1, +∞; > ∘ lim 0, 1 a, b 0bxx→+∞ a = , → +∞ a, >blog x ox x b 0γlog|x| a = a, >  log|x| = ,

→ −∞ a, >∘ balim 0, b 0 o|x| x b 0b|x|x→−∞ alog |x|γ∘ = γ ∈ 0, ∪ 1, +∞; > lim 0, 1 a, b 0b|x|x→−∞ a = , → −∞ γ ∈ 0, ∪ 1, +∞; >blog o|x| x 1 a, b 0|x|γ———————————————————————————————————————————————————————————————————————————————————————————————————&mdash

definitivamente per x p, valgono le seguentirelazioni:

sin gx∘ =  ∽ = + →lim 1 sin gx gx; sin gx gx ogx, x pgx x→px→p

tan gx∘ =  ∽ = + →lim 1 tan gx gx; tan gx gx ogx, x pgx x→px→p

−1 cos gx∘ =  − ∽ gx1 1 2lim 1 cos gx ;gx 2 2 2x→px→p gx 2= − + , →2cos gx 1 ogx x p2

arctan gx =  ∽∘ lim 1 arctan gx gx;gx x→px→p = + →arctan gx gx ogx, x p

arcsin gx∘ =  ∽lim 1 arcsin gx gx;gx x→px→p = + →arcsin gx gx ogx, x p

α α∘ + αgx = α ∈  +

αgflxffiffi ∽1/gflxffi 1/gflxffifl1 ffi1lim e , Rffi e ;x→px→p α+ αgflxffiffi = + →1/gflxffifl1 e ofl1ffi, x p+logfl1 gflxffiffiˆ = ％ + ∽lim 1 logfl1 gflxffiffi gflxffi;gflxffi x→px→p + = + →logfl1 gflxffiffi gflxffi oflgflxffiffi, x pfl1 +log gflxffiffi gflxffi1ˆ = fla > ≠ ％ fl1 + ∽alim , 0, a 1ffi log gflxffiffi ,alog a log agflxffi x→pfla>0,a≠1ffix→p gflxffifl1 + = + →log gflxffiffi oflgflxffiffi, x pa log afla>0,a≠1ffi−gflxffie 1ˆ = ％ − ∽ = + + →gflxffi gflxffilim 1 e 1 gflxffi; e 1 gflxffi oflgflxffiffi, x pgflxffi x→px→p −gflxffi 1aˆ = fla > ％ − ∽ fl1 ≠ >gflxffilim log a, 0ffi a 1 gflxffi log a, a 0ffi ;gflxffi x→px→p = + + → fl1 ≠ >gflxffia 1 gflxffi log a oflgflxffiffi, x p a 0ffiαfl1 + −gflxffiffi 1 α= α, flα ∈ ％