Калькулятор границь

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