Analisi matematica I - Limiti

Forme ind.

Limiti x -> 1⁺ (x/(√x-1) - (√x+2)/(x-1)) = ∞ - ∞= (x/(√x-1) · (√x+1)/(√x+1) - (√x+2)/(x-1)) = (x(√x+1)/(x-1) - (√x+2)/(x-1)) =((x√x + x - √x - 2)/(x-1)) = (-1/0⁺) = -∞

lim x -> +∞ (√x²+1 - x) = ∞ - ∞√x²+1 - x · (√x²+1 + x)/(√x²+1 + x) = (x²+1 - x²)/( √x²+1 + x) = 0 (0⁺)

lim x -> ∞ (x³-1/x²+1 + (-x²+2)) = ∞ - ∞= (x³-1 - x⁴ - x² + 2x² + 2)/(x²+1) = (-x⁴ + x³ + x² + 1)/(x²+1) = (x²(-1 + 1/x + 1/x² + 1/x⁴))/(x²(1 + 1/x²))= -∞

Forme ind.

lim_{x -> 1⁺} (x/(sqrt(x)-1) - (sqrt(x)+2)/(x-1)) = infty - infty= (x/(sqrt(x)-1) . (sqrt(x)+1)/(sqrt(x)+1) - (sqrt(x)+2)/(x-1)) = (x(sqrt(x)+1)/(x-1) - (sqrt(x)+2)/(x-1)) =((x sqrt(x)+x-sqrt(x)-2)/(x-1)) = (-1/0⁺) = -infty

lim_{x -> +infty} (sqrt(x²+1) - x) = infty - inftysqrt(x²+1) - x . (sqrt(x²+1)+x)/(sqrt(x²+1)+x) = (x²+1-x²)/(sqrt(x²+1)+x) = 0 (0⁺)

lim_{x -> infty} (x³-1)/(x²+1) + (-x²+2) = infty - infty= (x³-1-x⁴-x²+2x²+2)/(x²+1) = (-x⁴+x³+x²+1)/(x²+1) = (x²(-1+1/x+1/x²+1/x⁴))/(x²(1+1/x²))= -infty

Limiti di funzioni

lim x³ + x² - 3√x = 0

x > 0

lim x³ + x² - 3√x = ∞ - ∞

x → +∞ x³(1 + 1/x - 3√x/x³) = +∞

lim x³ + x² - 3√x = ?

x → -∞

lim 1/(2 - x) = 1

x → 1

lim 1/(2 - x) = ∞

x → 2

lim = 0^-

x → +∞

lim = 0⁺

x → -∞

lim (n - 1)/(n + 1) = n(1 - 1/n)/n(1 + 1/n) = 1

n → +∞

lim (2n - 6)/(n² + 1) = n(2 - 6/n)/n²(1 + 1/n²) = 0

n →∞

lim n/(n + (-1)ⁿ) = n/n(1 + (-1)ⁿ/n) = 1

n →∞

lim (n² - n)/(2n - 1) = n²(1 - 1/n)/n(2 - 1/n) = ∞

n →∞

lim (n² - 5n)/(n + 6n⁸) = n²(1 - 5n/n²)/n²(6 + 1/n) = 1/6

n →∞

Esercizi limiti successioni

(1) lim (eⁿ - 2ⁿ) = +infinito

n -> +infinito

eⁿ(1 - 2ⁿ/eⁿ) = +infinito

(2) lim (2ⁿ⁺¹ + 1)/(3ⁿ + 1) = 0

n -> +infinito

2ⁿ(2 + 1/2ⁿ)/3ⁿ(1 + 1/3ⁿ) = (2/3)ⁿ = 0

(3) lim ⁿ√2n = 1

n -> +infinito

(2n)^1/n = e^(1/n)ln(2n)

ln(2n)/n = 0

(4) lim (n - log n) = +infinito

n -> +infinito

n = 10^x

log n = x

n -> +infinito, x -> +infinito

lim (10^x - x) = x(10^x/x - 1) = +infinito

x -> +infinito

(5) lim (2ⁿ - 4ⁿ)/(3ⁿ - n!) = 0

n -> +infinito

2ⁿ/3ⁿ - 4ⁿ/3ⁿn!

2ⁿ/3ⁿ(1 - n!/3ⁿ) - 4ⁿ/3ⁿ(1 - n!/3ⁿ)

Esercizi limiti

a) lim_n→∞ (1 + 1/(3n))²ⁿ = ((1 + 1/(3n))³ⁿ)^2/3 = e^2/3

b) lim_n→∞ (sqrt(n) - n + n²)/(2n² - n^3/2 + 1) = (sqrt(n) - n + n²)/(2n² - n sqrt(n) + 1) = n²(sqrt(n)/n² - 1/n + 1)/(n²(2 - sqrt(n)/n + 1/n²)) = 1/2

c) lim_n→∞ (2ⁿ - 3ⁿ)/(1 + 3ⁿ) = 2ⁿ/(1 + 3ⁿ) - 3ⁿ/(1 + 3ⁿ) = 2ⁿ/(3ⁿ(1 + 1/3ⁿ)) - 3ⁿ/(3ⁿ(1 + 1/3ⁿ)) = -1

d) lim_n→∞ (2ⁿ + n²)/(3ⁿ + n³) = 2ⁿ/(3ⁿ + n³) + n²/(3ⁿ + n³) = 2ⁿ/(3ⁿ(1 + n³/3ⁿ)) + n²/(n³(3ⁿ/n³ + 1)) = 0

e) lim_n→∞ n log n/((n + 1)(n + 2)) = n log n/(n² + 3n + 2) = n log n/(n²(1 + 3/n + 2/n²)) = 0

g) lim_n→∞ (1 + log n)/(sqrt(n) - log n) = 1/(sqrt(n)(1 - log n/sqrt(n))) + log n/(log n(-1 + sqrt(n)/log n)) = 0

g) lim ( -1 )ⁿ n/(n²+1) = 0

n->oo

h) lim ( -1 )ⁿ (n²+1)/(n+1) = ?

n->oo

i) lim ⁿ√(2ⁿ+3ⁿ) = ⁿ√(3ⁿ(1+(2/3)ⁿ)) = ⁿ√3ⁿ . ⁿ√(1+(2/3)ⁿ) = 3

n->oo

l) lim ⁿ√(2n/(3n²+1)) = ⁿ√n . ⁿ√2 / ⁿ√(nⁿ(3+1/n²)) = ⁿ√2 / (ⁿ√n · ⁿ√(3+1/n²)) = 1

n->oo

ⁿ√n = n^1/n = e^{(1/n) ln n} = e^{ln n/n} = e⁰ = 1

m) lim n²(3ⁿ - 3^-n)/(4ⁿ + n²) = n²3ⁿ(1 - 3^-2n)/(4ⁿ(1 + n²/4ⁿ)) = n²3ⁿ/4ⁿ · (1 - 3^-2n)/(1 + n²/4ⁿ)

n->oo

n² · 3ⁿ/4ⁿ = n²/(4/3)ⁿ = 0

n) lim_n->∞ (n⁶ + log n + 3ⁿ) / (2ⁿ + n⁴ + log⁵ n)

n⁶ / (2ⁿ + n⁴ + log⁵ n) + log n / (2ⁿ + n⁴ + log⁵ n) + 3ⁿ / (2ⁿ + n⁴ + log⁵ n) = +∞

o) lim_n->∞ ((n + 3) / (n + 1))ⁿ = (1 + 2/(n+1))ⁿ = ((1 + 2/(n+1))^(n+1)/2)^2n/(n+1) = e²

p) lim_n->∞ ((n - 1) / n)ⁿ² = (1 - 1/n)ⁿ² = ((1 + 1/(-n))^-n) = e^-n = 1/eⁿ = 0

q) lim_n->∞ (n² + 1)ⁿ / n²ⁿ = ((n² + 1) / n²)ⁿ = ((1 + 1/n²)ⁿ²)^1/n = e^1/n = 1

r) lim_n->∞ log(n + 1) / log n = log[n(1 + 1/n)] / log n = log n + log(1 + 1/n) / log n = 1

1) lim_n->+∞ (ⁿ√3 - 1)ⁿ = 0^+∞ = 0

t) lim (nⁿlog n)_n->∞ = n^1/n . (log n)^1/n = (ln)^1/n . (log n)^1/n

e^{(1/n) ln n} . e^{ln(log n)/n} → 1

u) lim n²2^-sqrt(n)_n->∞ = n²/2^sqrt(n) = 0

v) lim (n^sqrt(n) - 2ⁿ)_n->∞

e^{sqrt(n) ln n} - e^{n ln 2} = -e^{n ln 2} (1 - e^{sqrt(n) ln n - n ln 2})

e^{sqrt(n) ln n - n ln 2} = e^{n((sqrt(n) ln n)/n - ln 2)} = e^-∞ = 0

Esercizi sulle successioni numeriche

1) lim (sqrt(n²+1) - (n²+1)/(n+1))     - ∞

n->∞

= ( sqrt(n²+1) · sqrt(n²+1)/sqrt(n²+1) - (n²+1)/(n+1) ) = ( (n²+1)/sqrt(n²+1) - (n²+1)/(n+1) )

= ( n³+n²+n+1 - n²sqrt(n²+1) - sqrt(n²+1) ) / (sqrt(n²+1)(n+1)) = 1

2) lim ln ⁿ√(1+eⁿ)     e > 1

n->∞

ln(1+eⁿ) = 1/n ln(1+eⁿ) < 1/n ln(e^1/n(e^1/n+1))

= 1/n ln eⁿ + 1/n ln(1/eⁿ + 1) = n/n ln e + 0 = ln e

3) lim 2^{sqrt((ln n)2+ln n2)} / (n²+1)

n->∞

(ln²n (1 + 2/ln n))^1/2(√((ln n)²+ln n²)) / ((ln²n + 2ln n))^1/2

ln n √1+2/√ln n

2/(n²+1) = 0

4) lim log₁₀((n² + 1)¹)

n → ∞

= (log_e(10)^ln(n)) · e^ln(n)/(n² + 1) = (10/e)^ln(n) · n/(n² + 1)

5) lim (1 + tan(1/n))ⁿ ~ (1 + 1/n)ⁿ = e

n → ∞

oppure (1 + tan(1/n))ⁿ = e^{n sin(1/n)/cos(1/n)} . n tg(1/n) = e¹ = e

6) lim arctan(n² - 1)

n → ∞

7) lim (1 + 1/(2n))^{3n + ln n} = (1 + 1/(2n))³ⁿ . (1 + 1/(2n))^{ln n}

n → ∞

= ((1 + 1/(2n))²ⁿ)^3/2 . (((1 + 1/(2n))²ⁿ)^{ln n/(2n)}→ e^3/2 . e^{ln n/(2n)} = e^3/2 . e⁰ = e^3/2

8)lim_{n -> oo} (2 + n²) / n³ cd 1 / n = 0

(2 / n³ + 1 / n) -> 1

2 / n³ cd 1 / n cd 2 / n³ cd 1 = 0

9)lim_{n -> oo} (1 + 2 / n³ + 1 / n)

((1 + 2 / n³)^1/n) cd (1 + 1 / n)^1/n

1 / n cd (2 / n³ + 1 / n) = 0

10)lim_{n -> oo} (sqrt(n+1) - sqrt(n-1))

sqrt(n)(sqrt(n+1) - sqrt(n-1) . (sqrt(n+1) + sqrt(n-1)) / (sqrt(n+1) + sqrt(n-1))) . sqrt(n)

((n+1 - n + 1) / (sqrt(n+1) + sqrt(n-1))) . sqrt(n) = 2 / (sqrt(n+1) + sqrt(n-1)) = 1

11) lim n² 2ⁿ/3ⁿ = n²(2/3)ⁿ = ∞ · 0

n→∞

n²/(3/2)ⁿ = 0

12) lim n² - sen n/(n²+n) = n²/(n²+n) - sen n/(n²+n) = 1

n→∞

13) lim (nⁿ₀ - 1)ⁿ = (0^1/n - 1)ⁿ = 0^∞ = 0

n→∞

14) lim (n³ + n² sen 1/n)/(n² + 1) = n³/(n²+1) + n² sen 1/n/(n²+1) = ∞

n→∞

15) lim (cos pi x)²ⁿ

n→∞

cos pi x ≤ 1

pi x ≠ 0

x ≠ 0

16) lim n² sen 1/n (ln n)²/(n+1)² = sen 1/n · ln²n/(n²+2n) = 0

n→∞

7)lim_n->∞ n[(n+2)^1/3 - n^1/3]

a = (n+2)^1/3    b = n^1/3

= n[(a-b)(a²+ab+b²)]/(a²+ab+b²) = n[(a-b)] = n[ka+k= 2n/(e^2/3 ln(n+2) + e^1/3 ln(n+2n) + e^2/3) = 0

8)lim_n->∞ (n/(n-1))^n/(n+1) = ((n+1)/(n-1))^n/(n+1)

= (1 + 1/(n-1))^n/(n+1) = (((1 + 1/(n-1))^(n-1))^{n/((n+1)(n-1))}) = e⁰ = 1

10)lim_n->∞ ((n²+n)/(n²-n+2))ⁿ = (n²+n-2n+2n+2-2)/(n²-n+2)ⁿ

= (1 + (2n-2)/(n²-n+2))ⁿ = (((1 + (2n-2)/(n²-n+2))^{(n2-n+2)/(2n-2)})^{n(2n-2)/(n2-n+2)})

= e^{2n2-2n)/(n2-n+2)} = e^{n2(2-2/n)/(n2(1-1/n+2/n2))} = e²