Limit Calculator
Calculate limits step by step
This free calculator will try to find the limit (two-sided or one-sided, including left and right) of the given function at the given point (including infinity), with steps shown.
Different techniques are used to handle limits (including indeterminate forms): limit laws, rewriting and simplifying, L'Hôpital's rule, rationalizing the denominator, taking natural logarithm, etc.
Solution
Your input: find $$$\lim_{x \to -\infty} \frac{x^{2} - 7}{x - 4}$$$
Multiply and divide by $$$x$$$:
$${\color{red}{\lim_{x \to -\infty} \frac{x^{2} - 7}{x - 4}}} = {\color{red}{\lim_{x \to -\infty} \frac{x \frac{x^{2} - 7}{x}}{x \frac{x - 4}{x}}}}$$
Divide:
$${\color{red}{\lim_{x \to -\infty} \frac{x \frac{x^{2} - 7}{x}}{x \frac{x - 4}{x}}}} = {\color{red}{\lim_{x \to -\infty} \frac{x - \frac{7}{x}}{1 - \frac{4}{x}}}}$$
The limit of the quotient is the quotient of limits:
$${\color{red}{\lim_{x \to -\infty} \frac{x - \frac{7}{x}}{1 - \frac{4}{x}}}} = {\color{red}{\frac{\lim_{x \to -\infty}\left(x - \frac{7}{x}\right)}{\lim_{x \to -\infty}\left(1 - \frac{4}{x}\right)}}}$$
The limit of a sum/difference is the sum/difference of limits:
$$\frac{{\color{red}{\lim_{x \to -\infty}\left(x - \frac{7}{x}\right)}}}{\lim_{x \to -\infty}\left(1 - \frac{4}{x}\right)} = \frac{{\color{red}{\left(- \lim_{x \to -\infty} \frac{7}{x} + \lim_{x \to -\infty} x\right)}}}{\lim_{x \to -\infty}\left(1 - \frac{4}{x}\right)}$$
Apply the constant multiple rule $$$\lim_{x \to -\infty} c f{\left(x \right)} = c \lim_{x \to -\infty} f{\left(x \right)}$$$ with $$$c=7$$$ and $$$f{\left(x \right)} = \frac{1}{x}$$$:
$$\frac{\lim_{x \to -\infty} x - {\color{red}{\lim_{x \to -\infty} \frac{7}{x}}}}{\lim_{x \to -\infty}\left(1 - \frac{4}{x}\right)} = \frac{\lim_{x \to -\infty} x - {\color{red}{\left(7 \lim_{x \to -\infty} \frac{1}{x}\right)}}}{\lim_{x \to -\infty}\left(1 - \frac{4}{x}\right)}$$
The limit of a quotient is the quotient of limits:
$$\frac{\lim_{x \to -\infty} x - 7 {\color{red}{\lim_{x \to -\infty} \frac{1}{x}}}}{\lim_{x \to -\infty}\left(1 - \frac{4}{x}\right)} = \frac{\lim_{x \to -\infty} x - 7 {\color{red}{\frac{\lim_{x \to -\infty} 1}{\lim_{x \to -\infty} x}}}}{\lim_{x \to -\infty}\left(1 - \frac{4}{x}\right)}$$
The limit of a constant is equal to the constant:
$$\frac{\lim_{x \to -\infty} x - \frac{7 {\color{red}{\lim_{x \to -\infty} 1}}}{\lim_{x \to -\infty} x}}{\lim_{x \to -\infty}\left(1 - \frac{4}{x}\right)} = \frac{\lim_{x \to -\infty} x - \frac{7 {\color{red}{1}}}{\lim_{x \to -\infty} x}}{\lim_{x \to -\infty}\left(1 - \frac{4}{x}\right)}$$
Constant divided by a very big number equals $$$0$$$:
$$\frac{\lim_{x \to -\infty} x - 7 {\color{red}{1 \frac{1}{\lim_{x \to -\infty} x}}}}{\lim_{x \to -\infty}\left(1 - \frac{4}{x}\right)} = \frac{\lim_{x \to -\infty} x - 7 {\color{red}{\left(0\right)}}}{\lim_{x \to -\infty}\left(1 - \frac{4}{x}\right)}$$
The function decreases without a bound:
$$\lim_{x \to -\infty} x = -\infty$$
The limit of a sum/difference is the sum/difference of limits:
$$- \infty {\color{red}{\lim_{x \to -\infty}\left(1 - \frac{4}{x}\right)}}^{-1} = - \infty {\color{red}{\left(\lim_{x \to -\infty} 1 - \lim_{x \to -\infty} \frac{4}{x}\right)}}^{-1}$$
The limit of a constant is equal to the constant:
$$- \infty \left(- \lim_{x \to -\infty} \frac{4}{x} + {\color{red}{\lim_{x \to -\infty} 1}}\right)^{-1} = - \infty \left(- \lim_{x \to -\infty} \frac{4}{x} + {\color{red}{1}}\right)^{-1}$$
Apply the constant multiple rule $$$\lim_{x \to -\infty} c f{\left(x \right)} = c \lim_{x \to -\infty} f{\left(x \right)}$$$ with $$$c=4$$$ and $$$f{\left(x \right)} = \frac{1}{x}$$$:
$$- \infty \left(1 - {\color{red}{\lim_{x \to -\infty} \frac{4}{x}}}\right)^{-1} = - \infty \left(1 - {\color{red}{\left(4 \lim_{x \to -\infty} \frac{1}{x}\right)}}\right)^{-1}$$
The limit of a quotient is the quotient of limits:
$$- \infty \left(1 - 4 {\color{red}{\lim_{x \to -\infty} \frac{1}{x}}}\right)^{-1} = - \infty \left(1 - 4 {\color{red}{\frac{\lim_{x \to -\infty} 1}{\lim_{x \to -\infty} x}}}\right)^{-1}$$
The limit of a constant is equal to the constant:
$$- \infty \left(1 - \frac{4 {\color{red}{\lim_{x \to -\infty} 1}}}{\lim_{x \to -\infty} x}\right)^{-1} = - \infty \left(1 - \frac{4 {\color{red}{1}}}{\lim_{x \to -\infty} x}\right)^{-1}$$
Constant divided by a very big number equals $$$0$$$:
$$- \infty \left(1 - 4 {\color{red}{1 \frac{1}{\lim_{x \to -\infty} x}}}\right)^{-1} = - \infty \left(1 - 4 {\color{red}{\left(0\right)}}\right)^{-1}$$
Therefore,
$$\lim_{x \to -\infty} \frac{x^{2} - 7}{x - 4} = -\infty$$
Answer: $$$\lim_{x \to -\infty} \frac{x^{2} - 7}{x - 4}=-\infty$$$