Calcolatore dei limiti

Questa calcolatrice gratuita cercherà di trovare il limite (bilaterale o unilaterale, compresi sinistra e destra) della funzione data nel punto dato (compreso l'infinito), con i passi indicati.

Vengono utilizzate diverse tecniche per gestire i limiti (comprese le forme indeterminate): leggi sui limiti, riscrittura e semplificazione, regola di L'Hôpital, razionalizzazione del denominatore, logaritmo naturale, ecc.

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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 decreases without a bound:

$\lim_{x \to -\infty} x^{3} = -\infty$

Therefore,

$\lim_{x \to -\infty}\left(x^{3} - 3 x^{2}\right) = -\infty$

Answer: $\lim_{x \to -\infty}\left(x^{3} - 3 x^{2}\right)=-\infty$

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