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Limietcalculator
Bereken limieten stap voor stap
Deze gratis rekenmachine probeert de limiet (tweezijdig of eenzijdig, links of rechts) van de gegeven functie in het gegeven punt (inclusief oneindig) te vinden, waarbij de stappen worden getoond.
Er worden verschillende technieken gebruikt om limieten te behandelen (waaronder onbepaalde vormen): limietwetten, herschrijven en vereenvoudigen, de regel van L'Hôpital, het rationaliseren van de noemer, het nemen van de natuurlijke logaritme, enz.
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: $$$\lim_{x \to \infty}\left(x^{3} - 3 x^{2}\right)=\infty$$$