Intorni

Es.m.122 p. 1377

y = ln ln ln |x-1|

D_f = { x ∈ R / |x-1| > 0 ^ ln |x-1| > 0 ^ ln ln |x-1| > 0 }

|x-1| > 0 => x ≠ 1

ln |x-1| > 0 => |x-1| > 1 => x-1 < -1 v x-1 > 1

ln (ln |x-1|) > 0 => ln |x-1| > 1 => |x-1| > e

|x-1| > e => x-1 < -e v x-1 > e

x < 1-e v x > 1+e

D_f = (-∞, 1-e) U (1+e, +∞)

Es. n. 123 p. 1377

y = arcsen log₂ (x - 1)/x

D_f = { x in IR / (x != 0) /\ (x - 1)/x > 0 /\ -1 ≤ log₂ (x - 1)/x ≤ 1 }

{ (x - 1)/x > 0

1/2 ≤ (x - 1)/x ≤ 2

=> { (x - 1)/x ≥ 1/2

(x - 1)/x ≤ 2

{ (2x - 2 - x)/2x ≥ 0

(x - 1 - 2x)/x ≤ 0

{ (x - 2)/2x ≥ 0

(-x - 1)/x ≤ 0

{ x < 0 V x > 2

x ≤ -1 V x > 0

x ≤ -1 V x ≥ 2

D_f = (-oo,-1] U [2,+oo)

Es. y = x² => x = +/- y y > 0

y = x + sen x

∃ f / y = f(x) , ∃ g / z = g(y)

? ∃ h / z = h(x) = g[f(x)] = g º f

C_f ⊂eacute; D_g

GLI INTERVALLI

• INTERVALLO CHIUSO [a,b] = {x in R / a x₀⁺

  ∀ε>0 ∃ I⁺_ε(x₀) / ∀x ∈ I⁺_ε(x₀) si ha |f(x)-l|<ε

  lim f(x)=l

  x->x₀^-

  ∀ε>0 ∃ I^-_ε(x₀) / ∀x ∈ I^-_ε(x₀) si ha |f(x)-l|<ε

  Es.m. 166 p. 1454

  lim (x²-2x)/|x| = -2

  x->0⁺

  D_f = { x ∈ R / |x| != 0 ⇒ x != 0 }

  D_f ⊆ R₀

  ∀ε>0 ∃ I⁺_ε(0) / ∀x ∈ I⁺_ε(0) si ha |(x²-2x)/|x| + 2| < ε

  |(x²-2x+2|x|)/|x|| < ε

  { x>0

  |x| < ε } U { x<0

  |(x²-4x)/(-x)| < ε

  { x>0

  -ε<x<ε } U { x<0

  |x(x-4)/(-x)| < ε

  Es. m. 98 p. 1450

  lim_x→-1 (1/2)^1+x = 1

  D_f = R

  Poiché -1 ∈ D_f f(x) → f(-1) = 1

  ∀ε > 0 ∃ I_ε(-1) / ∀x ∈ I_ε(-1) si ha |(1/2)^1+x - 1| < ε

  { (1/2)^x+1 - 1 < ε(1/2)^x+1 - 1 > -ε }

  { (1/2)^x+1 < 1+ε(1/2)^x+1 > 1-ε }

  { x+1 > log_1/2(1+ε)x+1 < log_1/2(1-ε) }

  { x > -1 + log_1/2(1+ε)x < -1 + log_1/2(1-ε) }

  -1 + log_1/2(1+ε)

  -1 + log_1/2(1-ε)

  I_ε(-1)

  Es.m.90 p.1450

  lim x -> 2

  log_1/3(3-x)=0

  Df = { x in R / 3-x > 0 => x < 3 } = (-infty, 3)

  2 in Df => f(x) -> f(2) = log_1/3 1 = 0

  For every epsilon > 0 there exists I_epsilon(2) / for every x in I_epsilon(2) one has |log_1/3(3-x)| < epsilon

  { log_1/3(3-x) < epsilon log_1/3(3-x) > -epsilon }

  { 3-x > (1/3)^epsilon 3-x < (1/3)^-epsilon }

  { x < 3 - (1/3)^epsilon x > 3 - 3^epsilon }

  3-3^epsilon    epsilon    3-(1/3)^epsilon

  I_epsilon(2)

  (1/3)^x

  3^x

  epsilon

  LIMITE INFINITO PER x CHE TENDE AD UN VALORE FINITO.

  lim f(x) = oo

  x->x₀

  -M

  M

  f(x)<-e V f(x)>e

  V M > 0 E I_M(x₀) / "x E I_M(x₀) - {x₀} si ha |f(x)|>M

  lim 1/sqrt(x²-3) = +oo

  x->sqrt(3)⁺

  D_f = (-oo,-sqrt(3)) U (sqrt(3),+oo)

  V M > 0 E I_M⁺(sqrt(3)) / "x E I_M⁺(sqrt(3)) si ha 1/sqrt(x²-3) > M

  => sqrt(x²-3) < 1/M => x²-3 < 1/M² => x² < 3 + 1/M²

  => -sqrt(3 + 1/M²) < x < sqrt(3 + 1/M²)

  I_M⁺(sqrt(3))

  I_M^-(sqrt(3))

  lim 1/sqrt(x²-3) = +oo

  x->-sqrt(3)^-

  LIMITE FINITO PER x TENDENTE ALL'INFINITO

  lim f(x) = l

  x -> oo

  V e > 0 E N_e > 0 / VX e Df com |x| > N_e si ha |f(x) - l| < e

  lim 1 / sqrt(x²-3) = 0

  x -> oo

  V e > 0 E N_e > 0 / VX e (-oo,-sqrt(3)) U (sqrt(3),+oo) com |x| > N_e si ha |1/sqrt(x²-3)| < e

  sqrt(x²-3) > 1/e 1/e² => x² > 3 + 1/e² => x < -sqrt(3 + 1/e²) v x > sqrt(3 + 1/e²)

  -sqrt(3 + 1/e²)

  sqrt(3 + 1/e²)

  -sqrt(3) sqrt(3)

  LIMITE INFINITO PER x TENDETE

  all' INFINITO.

  lim_{x -> oo} f(x) = oo

  V M > 0 E N_M > 0 / Vx e D_f come |x| > N_M sia |f(x)| > M