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Grenzwertrechner
Grenzwerte Schritt für Schritt berechnen
Dieser kostenlose Rechner versucht, den Grenzwert (zweiseitig oder einseitig, links- oder rechtsseitig) der gegebenen Funktion an der gegebenen Stelle (einschließlich im Unendlichen) zu bestimmen, mit angezeigten Rechenschritten.
Zur Behandlung von Grenzwerten (auch bei unbestimmten Formen) werden verschiedene Techniken eingesetzt: Grenzwertgesetze, Umformen und Vereinfachen, die Regel von L'Hôpital, Rationalisieren des Nenners, Bildung des natürlichen Logarithmus usw.
Solution
Your input: find $$$\lim_{x \to \infty} \frac{x + 1}{\sqrt[3]{x^{3} + 1}}$$$
Multiply and divide by $$$x$$$:
$${\color{red}{\lim_{x \to \infty} \frac{x + 1}{\sqrt[3]{x^{3} + 1}}}} = {\color{red}{\lim_{x \to \infty} \frac{x \frac{x + 1}{x}}{x \frac{\sqrt[3]{x^{3} + 1}}{x}}}}$$
Divide:
$${\color{red}{\lim_{x \to \infty} \frac{x \frac{x + 1}{x}}{x \frac{\sqrt[3]{x^{3} + 1}}{x}}}} = {\color{red}{\lim_{x \to \infty} \frac{1 + \frac{1}{x}}{\sqrt[3]{1 + \frac{1}{x^{3}}}}}}$$
The limit of the quotient is the quotient of limits:
$${\color{red}{\lim_{x \to \infty} \frac{1 + \frac{1}{x}}{\sqrt[3]{1 + \frac{1}{x^{3}}}}}} = {\color{red}{\frac{\lim_{x \to \infty}\left(1 + \frac{1}{x}\right)}{\lim_{x \to \infty} \sqrt[3]{1 + \frac{1}{x^{3}}}}}}$$
The limit of a sum/difference is the sum/difference of limits:
$$\frac{{\color{red}{\lim_{x \to \infty}\left(1 + \frac{1}{x}\right)}}}{\lim_{x \to \infty} \sqrt[3]{1 + \frac{1}{x^{3}}}} = \frac{{\color{red}{\left(\lim_{x \to \infty} 1 + \lim_{x \to \infty} \frac{1}{x}\right)}}}{\lim_{x \to \infty} \sqrt[3]{1 + \frac{1}{x^{3}}}}$$
The limit of a constant is equal to the constant:
$$\frac{\lim_{x \to \infty} \frac{1}{x} + {\color{red}{\lim_{x \to \infty} 1}}}{\lim_{x \to \infty} \sqrt[3]{1 + \frac{1}{x^{3}}}} = \frac{\lim_{x \to \infty} \frac{1}{x} + {\color{red}{1}}}{\lim_{x \to \infty} \sqrt[3]{1 + \frac{1}{x^{3}}}}$$
The limit of a quotient is the quotient of limits:
$$\frac{1 + {\color{red}{\lim_{x \to \infty} \frac{1}{x}}}}{\lim_{x \to \infty} \sqrt[3]{1 + \frac{1}{x^{3}}}} = \frac{1 + {\color{red}{\frac{\lim_{x \to \infty} 1}{\lim_{x \to \infty} x}}}}{\lim_{x \to \infty} \sqrt[3]{1 + \frac{1}{x^{3}}}}$$
The limit of a constant is equal to the constant:
$$\frac{1 + \frac{{\color{red}{\lim_{x \to \infty} 1}}}{\lim_{x \to \infty} x}}{\lim_{x \to \infty} \sqrt[3]{1 + \frac{1}{x^{3}}}} = \frac{1 + \frac{{\color{red}{1}}}{\lim_{x \to \infty} x}}{\lim_{x \to \infty} \sqrt[3]{1 + \frac{1}{x^{3}}}}$$
Constant divided by a very big number equals $$$0$$$:
$$\frac{1 + {\color{red}{1 \frac{1}{\lim_{x \to \infty} x}}}}{\lim_{x \to \infty} \sqrt[3]{1 + \frac{1}{x^{3}}}} = \frac{1 + {\color{red}{\left(0\right)}}}{\lim_{x \to \infty} \sqrt[3]{1 + \frac{1}{x^{3}}}}$$
Move the limit under the power:
$${\color{red}{\lim_{x \to \infty} \sqrt[3]{1 + \frac{1}{x^{3}}}}}^{-1} = {\color{red}{\sqrt[3]{\lim_{x \to \infty}\left(1 + \frac{1}{x^{3}}\right)}}}^{-1}$$
The limit of a sum/difference is the sum/difference of limits:
$$\frac{1}{\sqrt[3]{{\color{red}{\lim_{x \to \infty}\left(1 + \frac{1}{x^{3}}\right)}}}} = \frac{1}{\sqrt[3]{{\color{red}{\left(\lim_{x \to \infty} 1 + \lim_{x \to \infty} \frac{1}{x^{3}}\right)}}}}$$
The limit of a constant is equal to the constant:
$$\frac{1}{\sqrt[3]{\lim_{x \to \infty} \frac{1}{x^{3}} + {\color{red}{\lim_{x \to \infty} 1}}}} = \frac{1}{\sqrt[3]{\lim_{x \to \infty} \frac{1}{x^{3}} + {\color{red}{1}}}}$$
The limit of a quotient is the quotient of limits:
$$\frac{1}{\sqrt[3]{1 + {\color{red}{\lim_{x \to \infty} \frac{1}{x^{3}}}}}} = \frac{1}{\sqrt[3]{1 + {\color{red}{\frac{\lim_{x \to \infty} 1}{\lim_{x \to \infty} x^{3}}}}}}$$
The limit of a constant is equal to the constant:
$$\frac{1}{\sqrt[3]{1 + \frac{{\color{red}{\lim_{x \to \infty} 1}}}{\lim_{x \to \infty} x^{3}}}} = \frac{1}{\sqrt[3]{1 + \frac{{\color{red}{1}}}{\lim_{x \to \infty} x^{3}}}}$$
Constant divided by a very big number equals $$$0$$$:
$$\frac{1}{\sqrt[3]{1 + {\color{red}{1 \frac{1}{\lim_{x \to \infty} x^{3}}}}}} = \frac{1}{\sqrt[3]{1 + {\color{red}{\left(0\right)}}}}$$
Therefore,
$$\lim_{x \to \infty} \frac{x + 1}{\sqrt[3]{x^{3} + 1}} = 1$$
Answer: $$$\lim_{x \to \infty} \frac{x + 1}{\sqrt[3]{x^{3} + 1}}=1$$$