Prove d'esame: integrali e Limiti svolti

Limiti matematici

Limiti notevoli e loro calcolo

1) Lim_x→0 (cos x)^1/x = 1^∞

Lim_x→0 [f(x) - 1] log cos x => Lim_x→0 log cos x / x = 0/0

Lim_x→0 cos² x log cos x / cos² x x => Lim_x→0 log cos x / (λ·x²·x^-1) x = Lim_x→0 cos² x log cos x / (x²·x^-1) x

Div cambio variabile(cos x = 1 - t²) / (1 + t²) x→x => x→x

Lim_x→0 log (x²·x²_{lim x→0} ^{x (√2 x -1)}/_{√(x² (x²-1))} =>

_{lim x→0} ^{√x (√2 x -1)}/_√x =>

_{lim x→0} ^{(√2 x -1)}/_{4 • √x²} =>

_{lim x→0} √x² (1/2 x -1) = - ∞

Limiti all'infinito

1) Lim_x→∞ ^{√(1+secx) - √(1-secx)}/_√(1+6sinx)= Lim_x→∞ ^{√(1+secx) - √(1-secx)}/_√(2+6sinx) = ⁰/₀

Lim_x→∞ ^{√(4+x2) - √(4-x2)}/_√(x²-x²) = ⁰/_√2 ⇒

2) Lim_x→∞ ^{√x * secx}/_{x √(4+x²)}

3) Lim_x→∞ ^{secx - 1/x}/_√(4+x²) = ⁰/₀ ⇒ Lim_x→∞ secx - ¹/_x+1/x² = ¹/_{(x+x')√(4+x²)}

Stopci di Lim_x→∞ ¹/_{√(x(x+1)²)√(x+1)} = Lim_x→∞ ¹/_{√(x(x+...)√(x+1)√x}√(x²(x+1),...)_√(4+x²)) = e^±∞ = ∞

Line lx→∞ + per parziali di definizione x→∞ per valori di

Conclusi lim ¹/_∞ = ⊗

Altri limiti notevoli

1) Lim_x→0 (x⁴ + tg x - x) / (x⁴ + x sin x)= Lim_x→0 (tg x - x) / (x⁴ + x sin x)= Lim_x→0 (1 - (tg x / x)) / (x⁴ + x sin x)= Lim_x→0 ((tg x)^x - x) / (x⁴ + x sin x) Lim_x→0 ( - 1) / (x⁴ + x sin x) = 0

2) Lim_n→∞ 3ⁿ - 3²ⁿ / m^m= Lim_m→∞ (3ⁿ - 3_{ln m}) Lim_m→∞ 3ⁿ / m^m - 3 ln^m / m³) studia il Lim m^m = ∞ => Lim ^ln = ∞

Lim_m→∞ 1 / m³ = 0 quindi Lim_m→∞ m^m / 3^m = 0

2) Lim_x->+∞ Lim_x->+∞ 4x¹⁺⁵ Lim_x->+∞ f(x) = 0^∞ Lim_x->+∞ (3x¹⁺⁵ | log√(4x²⁺¹)) Lim_x->+∞ Lim_x->+∞ log (√(4x²+1)=0 :0:0 = > Lim_x->+∞ log (√(4x²+1) log Lim_x->+∞ log Lim_x->+∞ 1 / x¹⁺⁵ (log Lim_x->+∞ x¹⁺⁵ [0] Lim ^logx¹⁺⁵ = 0 Lim ^l0 Lim_x->+∞ 3(x+1) Lim_x->+∞ l⁰ = l

• Lim_x→1
• Lim_x→∞ f(x)= 1/3
• Lim_x→0