Limit Calculator

Your input: find $$$\lim_{x \to \infty}\left(x^{4} - 6 x^{2}\right)$$$

Multiply and divide by $$$x^{4}$$$:

$${\color{red}{\lim_{x \to \infty}\left(x^{4} - 6 x^{2}\right)}} = {\color{red}{\lim_{x \to \infty} x^{4} \frac{x^{4} - 6 x^{2}}{x^{4}}}}$$

Divide:

$${\color{red}{\lim_{x \to \infty} x^{4} \frac{x^{4} - 6 x^{2}}{x^{4}}}} = {\color{red}{\lim_{x \to \infty} x^{4} \left(1 - \frac{6}{x^{2}}\right)}}$$

The limit of a product/quotient is the product/quotient of limits:

$${\color{red}{\lim_{x \to \infty} x^{4} \left(1 - \frac{6}{x^{2}}\right)}} = {\color{red}{\lim_{x \to \infty} x^{4} \lim_{x \to \infty}\left(1 - \frac{6}{x^{2}}\right)}}$$

The limit of a sum/difference is the sum/difference of limits:

$$\lim_{x \to \infty} x^{4} {\color{red}{\lim_{x \to \infty}\left(1 - \frac{6}{x^{2}}\right)}} = \lim_{x \to \infty} x^{4} {\color{red}{\left(\lim_{x \to \infty} 1 - \lim_{x \to \infty} \frac{6}{x^{2}}\right)}}$$

The limit of a constant is equal to the constant:

$$\lim_{x \to \infty} x^{4} \left(- \lim_{x \to \infty} \frac{6}{x^{2}} + {\color{red}{\lim_{x \to \infty} 1}}\right) = \lim_{x \to \infty} x^{4} \left(- \lim_{x \to \infty} \frac{6}{x^{2}} + {\color{red}{1}}\right)$$

Apply the constant multiple rule $$$\lim_{x \to \infty} c f{\left(x \right)} = c \lim_{x \to \infty} f{\left(x \right)}$$$ with $$$c=6$$$ and $$$f{\left(x \right)} = \frac{1}{x^{2}}$$$:

$$\lim_{x \to \infty} x^{4} \left(1 - {\color{red}{\lim_{x \to \infty} \frac{6}{x^{2}}}}\right) = \lim_{x \to \infty} x^{4} \left(1 - {\color{red}{\left(6 \lim_{x \to \infty} \frac{1}{x^{2}}\right)}}\right)$$

The limit of a quotient is the quotient of limits:

$$\lim_{x \to \infty} x^{4} \left(1 - 6 {\color{red}{\lim_{x \to \infty} \frac{1}{x^{2}}}}\right) = \lim_{x \to \infty} x^{4} \left(1 - 6 {\color{red}{\frac{\lim_{x \to \infty} 1}{\lim_{x \to \infty} x^{2}}}}\right)$$

The limit of a constant is equal to the constant:

$$\lim_{x \to \infty} x^{4} \left(1 - \frac{6 {\color{red}{\lim_{x \to \infty} 1}}}{\lim_{x \to \infty} x^{2}}\right) = \lim_{x \to \infty} x^{4} \left(1 - \frac{6 {\color{red}{1}}}{\lim_{x \to \infty} x^{2}}\right)$$

Constant divided by a very big number equals $$$0$$$:

$$\lim_{x \to \infty} x^{4} \left(1 - 6 {\color{red}{1 \frac{1}{\lim_{x \to \infty} x^{2}}}}\right) = \lim_{x \to \infty} x^{4} \left(1 - 6 {\color{red}{\left(0\right)}}\right)$$

The function grows without a bound:

$$\lim_{x \to \infty} x^{4} = \infty$$

Therefore,

$$\lim_{x \to \infty}\left(x^{4} - 6 x^{2}\right) = \infty$$

Answer: $$$\lim_{x \to \infty}\left(x^{4} - 6 x^{2}\right)=\infty$$$