Indice
Proprietà dei limiti
Se[math]\lim\limits_{x \to x_0} f(x) = l_1 \in \mathbb{R}[/math]
e [math]\lim\limits_{x \to x_0} g(x) = l_2 \in \mathbb{R}[/math]
, allora[math]\lim\limits_{x \to x_0} c \cdot f(x) = c \cdot l_1[/math]
, per ogni [math]c \in \mathbb{R}[/math]
[math]\lim\limits_{x \to x_0} f(x) + g(x) = l_1 + l_2[/math]
[math]\lim\limits_{x \to x_0} f(x) - g(x) = l_1 - l_2[/math]
[math]\lim\limits_{x \to x_0} f(x) \cdot g(x) = l_1 \cdot l_2[/math]
[math]\lim\limits_{x \to x_0} \frac{1}{f(x)} = \frac{1}{l_1}[/math]
, se [math]l_1 \mbox{ se } 0[/math]
[math]\lim\limits_{x \to x_0} \frac{f(x)}{g(x)} = \frac{l_1}{l_2}[/math]
, se [math]l_2 \mbox{ se } 0[/math]
Se [math]\lim\limits_{x \to x_0} f(x) = l \in \mathbb{R}[/math]
e [math]\lim\limits_{x \to x_0} g(x) = \pm \infty[/math]
, allora
[math]\lim\limits_{x \to x_0} f(x) + g(x) = \pm\infty[/math]
[math]\lim\limits_{x \to x_0} f(x) - g(x) = \mp\infty[/math]
Se [math]\lim\limits_{x \to x_0} f(x) = \lim\limits_{x \to x_0} g(x) = \pm \infty[/math]
, allora[math]\lim\limits_{x \to x_0} f(x) + g(x) = \pm \infty[/math]
[math]\lim\limits_{x \to x_0} f(x) \cdot g(x) = +\infty[/math]
Se [math]lim_{x \to x_0} f(x) = l \in \mathbb{R} \setminus {0}[/math]
e [math]\lim\limits_{x \to x_0} g(x) = \pm \infty[/math]
, allora
[math]\lim\limits_{x \to x_0} f(x) \cdot g(x) = {(\pm \infty, \mbox{" se "} l > 0),(\mp \infty, \mbox{" se "} l
[math]\lim\limits_{x \to x_0} \frac{f(x)}{g(x)} = 0[/math]
Se [math]\lim\limits_{x \to x_0} f(x)[/math]
non esiste, ma [math]f(x)[/math]
è una funzione limitata, e se [math]\lim\limits_{x \to x_0} g(x) = 0[/math]
, allora
[math]\lim\limits_{x \to x_0} f(x) \cdot g(x) = 0[/math]
Se [math]\lim\limits_{x \to x_0} f(x)[/math]
non esiste, ma [math]f(x)[/math]
è una funzione limitata, e se [math]\lim\limits_{x \to x_0} g(x) = \pm \infty[/math]
, allora
[math]\lim\limits_{x \to x_0} f(x) + g(x) = \pm \infty[/math]
[math]\lim\limits_{x \to x_0} \frac{f(x)}{g(x)} = 0[/math]
Tavola dei limiti notevoli
Razionali
[math]\lim\limits_{x \to \pm \infty} \frac{a_n x^n + a_{n-1} x^{n-1} + \ldots + a_0}{b_m x^m + b_{m-1} x^{m-1} + \ldots + b_0} = {(+\infty, \mbox{" se "} n > m \mbox{" e "} \frac{a_n}{b_m} > 0),(-\infty, \mbox{" se "} n > m \mbox{" e "} \frac{a_n}{b_m}
Esponenziali e logaritmici
|
[math]\lim\limits_{x \to +\infty} (1 + \frac{1}{x})^x = e[/math] |
[math]\lim\limits_{x \to -\infty} (1 + \frac{1}{x})^x = e[/math] |
[math]\lim\limits_{x \to \pm\infty} (1 + \frac{a}{x})^{x} = e^{a}[/math] |
|
[math]\lim\limits_{x \to \pm\infty} (1 + \frac{a}{x})^{nx} = e^{na}[/math] |
[math]\lim\limits_{x \to \pm\infty} (1 - \frac{1}{x})^x = \frac{1}{e}[/math] |
[math]\lim\limits_{x \to 0} (1 + ax)^{\frac{1}{x}} = e^a[/math] |
[math]\lim\limits_{x \to 0} \log_{a} ((1 + x)^{\frac{1}{x}}) = \frac{1}{\log_{e}(a)}[/math] |
[math]\lim\limits_{x \to 0} \frac{\log_{a} (1 + x)}{x} = \frac{1}{\log_{e}(a)}[/math] |
[math]\lim\limits_{x \to 0} \frac{a^x - 1}{x} = \ln(a)[/math] , [math]\forall a \in \mathbb{R}^+[/math]
|
[math]\lim\limits_{x \to 0} \frac{(1+x)^a - 1}{x} = a[/math] |
[math]\lim\limits_{x \to 0} \frac{(1+x)^a - 1}{ax} = 1[/math] |
[math]\lim\limits_lim_{x \to 0} x^b log_{a}(x) = 0[/math] , [math]\forall b \in \mathbb{R}^+[/math] |
|
[math]\lim\limits_{x \to 0} \frac{\log_{a}(x)}{x^b} = +\infty[/math] , [math]\forall b \in \mathbb{R}^+[/math] , con [math]0
|
[math]\lim\limits_{x \to 0} \frac{\log_{a}(x)}{x^b} = -\infty[/math] , [math]\forall b \in \mathbb{R}^+[/math] , con [math]a > 1[/math]
|
[math]\lim\limits_{x \to +\infty} a^x = 0[/math] , [math]\forall a \in (0,1)[/math]
|
|
[math]\lim\limits_{x \to +\infty} a^x = +\infty[/math] , [math]\forall a \in (1, +\infty)[/math] |
[math]\lim\limits_{x \to -\infty} a^x = +\infty[/math] , [math]\forall a \in (0,1)[/math] |
[math]\lim\limits_{x \to -\infty} a^x = 0[/math] , [math]\forall a \in (1, +\infty)[/math] |
|
[math]\lim\limits_{x \to +\infty} x^b a^x = \lim\limits_{x \to +\infty} a^x[/math] , [math]\forall b \in \mathbb{R}^+[/math] , [math]\forall a \in \mathbb{R}^+ \setminus {1}[/math]
|
[math]\lim\limits_{x \to -\infty} |x|^b a^x = \lim\limits_{x \to -infty} a^x[/math] , [math]\forall b \in \mathbb{R}^+[/math]
|
[math]\lim\limits_{x \to +\infty} \frac{a^x}{x^b} = \lim\limits_{x \to +\infty} a^x[/math] , [math]\forall b \in \mathbb{R}^+[/math] , [math]\forall a \in \mathbb{R}^+ \setminus {1}[/math]
|
|
[math]\lim\limits_{x \to +\infty} \frac{x^b}{a^x} = \lim\limits_{x \to -\infty} a^x[/math] , [math]\forall b \in \mathbb{R}^+[/math] , [math]\forall a \in \mathbb{R}^+ \setminus {1}[/math] |
[math]\lim\limits_{x \to -\infty} e^x x^b = 0[/math] , [math]\forall b \in \mathbb{R}^+[/math] |
Goniometrici e iperbolici
|
[math]\lim\limits_{x \to 0} \frac{\sin(x)}{x} = 1[/math] |
[math]\lim\limits_{x \to 0} \frac{\sin(ax)}{bx} = \frac{a}{b}[/math] |
[math]\lim\limits_{x \to 0} \frac{\sin(ax)}{\sin(bx)} = \frac{a}{b}[/math] |
|
[math]\lim\limits_{x \to 0} \frac{\tan(x)}{x} = 1[/math] |
[math]\lim\limits_{x \to 0} \frac{\tan(ax)}{bx} = \frac{a}{b}[/math] |
[math]\lim\limits_{x \to 0} \frac{\tan(ax)}{\tan\(bx)} = \frac{a}{b}[/math] |
[math]\lim\limits_{x \to 0} \frac{1 - \cos(x)}{x} = 0[/math] |
[math]\lim\limits_{x \to 0} \frac{1 - \cos(x)}{x^2} = \frac{1}{2}[/math] |
[math]\lim\limits_{x \to 0} \frac{\arcsin(x)}{x} = 1[/math] |
[math]\lim\limits_{x \to 0} \frac{\arcsin(ax)}{bx} = frac{a}{b}[/math] |
[math]\lim\limits_{x \to 0} \frac{\arcsin(ax)}{\arcsin(bx)} = \frac{a}{b}[/math] |
[math]\lim\limits_{x \to 0} \frac{\arctan(x)}{x} = 1[/math] |
|
[math]\lim\limits_{x \to 0} \frac{\arctan(ax)}{bx} = \frac{a}{b}[/math] |
[math]\lim\limits_{x \to 0} \frac{\arctan(ax)}{\arctan(bx)} = \frac{a}{b}[/math] |
[math]\lim\limits_{x \to 0} \frac{\sinh(x)}{x} = 1[/math] |
|
[math]\lim\limits_{x \to 0} \frac{\sinh(x)}{x} = 1[/math] |
[math]\lim\limits_{x \to 0} \frac{\tanh(x)}{x} = 1[/math] |
[math]\lim\limits_{x \to 0} \frac{\tanh(x)}{x} = 1[/math] |
|
[math]\lim\limits_{x \to 0} \frac{x - \sin(x)}{x^3} = \frac{1}{6}[/math] |
[math]\lim\limits_{x \to 0} \frac{x - \arctan(x)}{x^3} = \frac{1}{3}[/math] |
Link utili
http://it.wikipedia.org/wiki/Tavola_dei_limiti_notevoli
Accedi a tutti gli appunti
Tutor AI: studia meglio e in meno tempo
