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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 $$$\lim_{x \to \infty}\left(x^{4} - x^{3} - 27 x^{2} + 25 x + 50\right)$$$
Multiply and divide by $$$x^{4}$$$:
$${\color{red}{\lim_{x \to \infty}\left(x^{4} - x^{3} - 27 x^{2} + 25 x + 50\right)}} = {\color{red}{\lim_{x \to \infty} x^{4} \frac{x^{4} - x^{3} - 27 x^{2} + 25 x + 50}{x^{4}}}}$$
Divide:
$${\color{red}{\lim_{x \to \infty} x^{4} \frac{x^{4} - x^{3} - 27 x^{2} + 25 x + 50}{x^{4}}}} = {\color{red}{\lim_{x \to \infty} x^{4} \left(1 - \frac{1}{x} - \frac{27}{x^{2}} + \frac{25}{x^{3}} + \frac{50}{x^{4}}\right)}}$$
The limit of a product/quotient is the product/quotient of limits:
$${\color{red}{\lim_{x \to \infty} x^{4} \left(1 - \frac{1}{x} - \frac{27}{x^{2}} + \frac{25}{x^{3}} + \frac{50}{x^{4}}\right)}} = {\color{red}{\lim_{x \to \infty} x^{4} \lim_{x \to \infty}\left(1 - \frac{1}{x} - \frac{27}{x^{2}} + \frac{25}{x^{3}} + \frac{50}{x^{4}}\right)}}$$
The limit of a sum/difference is the sum/difference of limits:
$$\lim_{x \to \infty} x^{4} {\color{red}{\lim_{x \to \infty}\left(1 - \frac{1}{x} - \frac{27}{x^{2}} + \frac{25}{x^{3}} + \frac{50}{x^{4}}\right)}} = \lim_{x \to \infty} x^{4} {\color{red}{\left(\lim_{x \to \infty} 1 + \lim_{x \to \infty} \frac{50}{x^{4}} + \lim_{x \to \infty} \frac{25}{x^{3}} - \lim_{x \to \infty} \frac{27}{x^{2}} - \lim_{x \to \infty} \frac{1}{x}\right)}}$$
The limit of a constant is equal to the constant:
$$\lim_{x \to \infty} x^{4} \left(\lim_{x \to \infty} \frac{50}{x^{4}} + \lim_{x \to \infty} \frac{25}{x^{3}} - \lim_{x \to \infty} \frac{27}{x^{2}} - \lim_{x \to \infty} \frac{1}{x} + {\color{red}{\lim_{x \to \infty} 1}}\right) = \lim_{x \to \infty} x^{4} \left(\lim_{x \to \infty} \frac{50}{x^{4}} + \lim_{x \to \infty} \frac{25}{x^{3}} - \lim_{x \to \infty} \frac{27}{x^{2}} - \lim_{x \to \infty} \frac{1}{x} + {\color{red}{1}}\right)$$
The limit of a quotient is the quotient of limits:
$$\lim_{x \to \infty} x^{4} \left(\lim_{x \to \infty} \frac{50}{x^{4}} + \lim_{x \to \infty} \frac{25}{x^{3}} - \lim_{x \to \infty} \frac{27}{x^{2}} + 1 - {\color{red}{\lim_{x \to \infty} \frac{1}{x}}}\right) = \lim_{x \to \infty} x^{4} \left(\lim_{x \to \infty} \frac{50}{x^{4}} + \lim_{x \to \infty} \frac{25}{x^{3}} - \lim_{x \to \infty} \frac{27}{x^{2}} + 1 - {\color{red}{\frac{\lim_{x \to \infty} 1}{\lim_{x \to \infty} x}}}\right)$$
The limit of a constant is equal to the constant:
$$\lim_{x \to \infty} x^{4} \left(\lim_{x \to \infty} \frac{50}{x^{4}} + \lim_{x \to \infty} \frac{25}{x^{3}} - \lim_{x \to \infty} \frac{27}{x^{2}} + 1 - \frac{{\color{red}{\lim_{x \to \infty} 1}}}{\lim_{x \to \infty} x}\right) = \lim_{x \to \infty} x^{4} \left(\lim_{x \to \infty} \frac{50}{x^{4}} + \lim_{x \to \infty} \frac{25}{x^{3}} - \lim_{x \to \infty} \frac{27}{x^{2}} + 1 - \frac{{\color{red}{1}}}{\lim_{x \to \infty} x}\right)$$
Constant divided by a very big number equals $$$0$$$:
$$\lim_{x \to \infty} x^{4} \left(\lim_{x \to \infty} \frac{50}{x^{4}} + \lim_{x \to \infty} \frac{25}{x^{3}} - \lim_{x \to \infty} \frac{27}{x^{2}} + 1 - {\color{red}{1 \frac{1}{\lim_{x \to \infty} x}}}\right) = \lim_{x \to \infty} x^{4} \left(\lim_{x \to \infty} \frac{50}{x^{4}} + \lim_{x \to \infty} \frac{25}{x^{3}} - \lim_{x \to \infty} \frac{27}{x^{2}} + 1 - {\color{red}{\left(0\right)}}\right)$$
Apply the constant multiple rule $$$\lim_{x \to \infty} c f{\left(x \right)} = c \lim_{x \to \infty} f{\left(x \right)}$$$ with $$$c=27$$$ and $$$f{\left(x \right)} = \frac{1}{x^{2}}$$$:
$$\lim_{x \to \infty} x^{4} \left(\lim_{x \to \infty} \frac{50}{x^{4}} + \lim_{x \to \infty} \frac{25}{x^{3}} + 1 - {\color{red}{\lim_{x \to \infty} \frac{27}{x^{2}}}}\right) = \lim_{x \to \infty} x^{4} \left(\lim_{x \to \infty} \frac{50}{x^{4}} + \lim_{x \to \infty} \frac{25}{x^{3}} + 1 - {\color{red}{\left(27 \lim_{x \to \infty} \frac{1}{x^{2}}\right)}}\right)$$
The limit of a quotient is the quotient of limits:
$$\lim_{x \to \infty} x^{4} \left(\lim_{x \to \infty} \frac{50}{x^{4}} + \lim_{x \to \infty} \frac{25}{x^{3}} + 1 - 27 {\color{red}{\lim_{x \to \infty} \frac{1}{x^{2}}}}\right) = \lim_{x \to \infty} x^{4} \left(\lim_{x \to \infty} \frac{50}{x^{4}} + \lim_{x \to \infty} \frac{25}{x^{3}} + 1 - 27 {\color{red}{\frac{\lim_{x \to \infty} 1}{\lim_{x \to \infty} x^{2}}}}\right)$$
The limit of a constant is equal to the constant:
$$\lim_{x \to \infty} x^{4} \left(\lim_{x \to \infty} \frac{50}{x^{4}} + \lim_{x \to \infty} \frac{25}{x^{3}} + 1 - \frac{27 {\color{red}{\lim_{x \to \infty} 1}}}{\lim_{x \to \infty} x^{2}}\right) = \lim_{x \to \infty} x^{4} \left(\lim_{x \to \infty} \frac{50}{x^{4}} + \lim_{x \to \infty} \frac{25}{x^{3}} + 1 - \frac{27 {\color{red}{1}}}{\lim_{x \to \infty} x^{2}}\right)$$
Constant divided by a very big number equals $$$0$$$:
$$\lim_{x \to \infty} x^{4} \left(\lim_{x \to \infty} \frac{50}{x^{4}} + \lim_{x \to \infty} \frac{25}{x^{3}} + 1 - 27 {\color{red}{1 \frac{1}{\lim_{x \to \infty} x^{2}}}}\right) = \lim_{x \to \infty} x^{4} \left(\lim_{x \to \infty} \frac{50}{x^{4}} + \lim_{x \to \infty} \frac{25}{x^{3}} + 1 - 27 {\color{red}{\left(0\right)}}\right)$$
Apply the constant multiple rule $$$\lim_{x \to \infty} c f{\left(x \right)} = c \lim_{x \to \infty} f{\left(x \right)}$$$ with $$$c=25$$$ and $$$f{\left(x \right)} = \frac{1}{x^{3}}$$$:
$$\lim_{x \to \infty} x^{4} \left(\lim_{x \to \infty} \frac{50}{x^{4}} + 1 + {\color{red}{\lim_{x \to \infty} \frac{25}{x^{3}}}}\right) = \lim_{x \to \infty} x^{4} \left(\lim_{x \to \infty} \frac{50}{x^{4}} + 1 + {\color{red}{\left(25 \lim_{x \to \infty} \frac{1}{x^{3}}\right)}}\right)$$
The limit of a quotient is the quotient of limits:
$$\lim_{x \to \infty} x^{4} \left(\lim_{x \to \infty} \frac{50}{x^{4}} + 1 + 25 {\color{red}{\lim_{x \to \infty} \frac{1}{x^{3}}}}\right) = \lim_{x \to \infty} x^{4} \left(\lim_{x \to \infty} \frac{50}{x^{4}} + 1 + 25 {\color{red}{\frac{\lim_{x \to \infty} 1}{\lim_{x \to \infty} x^{3}}}}\right)$$
The limit of a constant is equal to the constant:
$$\lim_{x \to \infty} x^{4} \left(\lim_{x \to \infty} \frac{50}{x^{4}} + 1 + \frac{25 {\color{red}{\lim_{x \to \infty} 1}}}{\lim_{x \to \infty} x^{3}}\right) = \lim_{x \to \infty} x^{4} \left(\lim_{x \to \infty} \frac{50}{x^{4}} + 1 + \frac{25 {\color{red}{1}}}{\lim_{x \to \infty} x^{3}}\right)$$
Constant divided by a very big number equals $$$0$$$:
$$\lim_{x \to \infty} x^{4} \left(\lim_{x \to \infty} \frac{50}{x^{4}} + 1 + 25 {\color{red}{1 \frac{1}{\lim_{x \to \infty} x^{3}}}}\right) = \lim_{x \to \infty} x^{4} \left(\lim_{x \to \infty} \frac{50}{x^{4}} + 1 + 25 {\color{red}{\left(0\right)}}\right)$$
Apply the constant multiple rule $$$\lim_{x \to \infty} c f{\left(x \right)} = c \lim_{x \to \infty} f{\left(x \right)}$$$ with $$$c=50$$$ and $$$f{\left(x \right)} = \frac{1}{x^{4}}$$$:
$$\lim_{x \to \infty} x^{4} \left(1 + {\color{red}{\lim_{x \to \infty} \frac{50}{x^{4}}}}\right) = \lim_{x \to \infty} x^{4} \left(1 + {\color{red}{\left(50 \lim_{x \to \infty} \frac{1}{x^{4}}\right)}}\right)$$
The limit of a quotient is the quotient of limits:
$$\lim_{x \to \infty} x^{4} \left(1 + 50 {\color{red}{\lim_{x \to \infty} \frac{1}{x^{4}}}}\right) = \lim_{x \to \infty} x^{4} \left(1 + 50 {\color{red}{\frac{\lim_{x \to \infty} 1}{\lim_{x \to \infty} x^{4}}}}\right)$$
The limit of a constant is equal to the constant:
$$\lim_{x \to \infty} x^{4} \left(1 + \frac{50 {\color{red}{\lim_{x \to \infty} 1}}}{\lim_{x \to \infty} x^{4}}\right) = \lim_{x \to \infty} x^{4} \left(1 + \frac{50 {\color{red}{1}}}{\lim_{x \to \infty} x^{4}}\right)$$
Constant divided by a very big number equals $$$0$$$:
$$\lim_{x \to \infty} x^{4} \left(1 + 50 {\color{red}{1 \frac{1}{\lim_{x \to \infty} x^{4}}}}\right) = \lim_{x \to \infty} x^{4} \left(1 + 50 {\color{red}{\left(0\right)}}\right)$$
The function grows without a bound:
$$\lim_{x \to \infty} x^{4} = \infty$$
Therefore,
$$\lim_{x \to \infty}\left(x^{4} - x^{3} - 27 x^{2} + 25 x + 50\right) = \infty$$
Answer: $$$\lim_{x \to \infty}\left(x^{4} - x^{3} - 27 x^{2} + 25 x + 50\right)=\infty$$$