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Calcolatore di limiti
Calcola i limiti passo dopo passo
Questo calcolatore gratuito proverà a trovare il limite (bilatero o unilatero, inclusi quello sinistro e quello destro) della funzione data nel punto indicato (incluso l'infinito), con i passaggi mostrati.
Per gestire i limiti (incluse le forme indeterminate) si utilizzano tecniche diverse: proprietà dei limiti, riscritture e semplificazioni, regola di De l'Hôpital, razionalizzazione del denominatore, applicazione del logaritmo naturale, ecc.
Solution
Your input: find $$$\lim_{x \to \infty}\left(x^{3} - 3 x^{2}\right)$$$
Multiply and divide by $$$x^{3}$$$:
$${\color{red}{\lim_{x \to \infty}\left(x^{3} - 3 x^{2}\right)}} = {\color{red}{\lim_{x \to \infty} x^{3} \frac{x^{3} - 3 x^{2}}{x^{3}}}}$$
Divide:
$${\color{red}{\lim_{x \to \infty} x^{3} \frac{x^{3} - 3 x^{2}}{x^{3}}}} = {\color{red}{\lim_{x \to \infty} x^{3} \left(1 - \frac{3}{x}\right)}}$$
The limit of a product/quotient is the product/quotient of limits:
$${\color{red}{\lim_{x \to \infty} x^{3} \left(1 - \frac{3}{x}\right)}} = {\color{red}{\lim_{x \to \infty} x^{3} \lim_{x \to \infty}\left(1 - \frac{3}{x}\right)}}$$
The limit of a sum/difference is the sum/difference of limits:
$$\lim_{x \to \infty} x^{3} {\color{red}{\lim_{x \to \infty}\left(1 - \frac{3}{x}\right)}} = \lim_{x \to \infty} x^{3} {\color{red}{\left(\lim_{x \to \infty} 1 - \lim_{x \to \infty} \frac{3}{x}\right)}}$$
The limit of a constant is equal to the constant:
$$\lim_{x \to \infty} x^{3} \left(- \lim_{x \to \infty} \frac{3}{x} + {\color{red}{\lim_{x \to \infty} 1}}\right) = \lim_{x \to \infty} x^{3} \left(- \lim_{x \to \infty} \frac{3}{x} + {\color{red}{1}}\right)$$
Apply the constant multiple rule $$$\lim_{x \to \infty} c f{\left(x \right)} = c \lim_{x \to \infty} f{\left(x \right)}$$$ with $$$c=3$$$ and $$$f{\left(x \right)} = \frac{1}{x}$$$:
$$\lim_{x \to \infty} x^{3} \left(1 - {\color{red}{\lim_{x \to \infty} \frac{3}{x}}}\right) = \lim_{x \to \infty} x^{3} \left(1 - {\color{red}{\left(3 \lim_{x \to \infty} \frac{1}{x}\right)}}\right)$$
The limit of a quotient is the quotient of limits:
$$\lim_{x \to \infty} x^{3} \left(1 - 3 {\color{red}{\lim_{x \to \infty} \frac{1}{x}}}\right) = \lim_{x \to \infty} x^{3} \left(1 - 3 {\color{red}{\frac{\lim_{x \to \infty} 1}{\lim_{x \to \infty} x}}}\right)$$
The limit of a constant is equal to the constant:
$$\lim_{x \to \infty} x^{3} \left(1 - \frac{3 {\color{red}{\lim_{x \to \infty} 1}}}{\lim_{x \to \infty} x}\right) = \lim_{x \to \infty} x^{3} \left(1 - \frac{3 {\color{red}{1}}}{\lim_{x \to \infty} x}\right)$$
Constant divided by a very big number equals $$$0$$$:
$$\lim_{x \to \infty} x^{3} \left(1 - 3 {\color{red}{1 \frac{1}{\lim_{x \to \infty} x}}}\right) = \lim_{x \to \infty} x^{3} \left(1 - 3 {\color{red}{\left(0\right)}}\right)$$
The function grows without a bound:
$$\lim_{x \to \infty} x^{3} = \infty$$
Therefore,
$$\lim_{x \to \infty}\left(x^{3} - 3 x^{2}\right) = \infty$$
Answer: $$$\lim_{x \to \infty}\left(x^{3} - 3 x^{2}\right)=\infty$$$