Grenzwert-Rechner

Dieser kostenlose Rechner wird versuchen, den Grenzwert (zweiseitig oder einseitig, einschließlich links und rechts) der gegebenen Funktion am gegebenen Punkt (einschließlich unendlich) zu finden, wobei die Schritte angezeigt werden.

Es werden verschiedene Techniken zur Behandlung von Grenzwerten (einschließlich unbestimmter Formen) verwendet: Grenzwertgesetze, Umschreiben und Vereinfachen, L'Hôpitalsche Regel, Rationalisierung des Nenners, natürlicher Logarithmus usw.

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$\displaystyle{\large{\infty}}$ , type inf.

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

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Solution