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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 $$$\lim_{x \to \infty} \frac{e^{x}}{x^{6}}$$$
Since we have an indeterminate form of type $$$\frac{\infty}{\infty}$$$, we can apply the l'Hopital's rule:
$${\color{red}{\lim_{x \to \infty} \frac{e^{x}}{x^{6}}}} = {\color{red}{\lim_{x \to \infty} \frac{\frac{d}{dx}\left(e^{x}\right)}{\frac{d}{dx}\left(x^{6}\right)}}}$$
For steps, see derivative calculator.
$${\color{red}{\lim_{x \to \infty} \frac{\frac{d}{dx}\left(e^{x}\right)}{\frac{d}{dx}\left(x^{6}\right)}}} = {\color{red}{\lim_{x \to \infty} \frac{e^{x}}{6 x^{5}}}}$$
Apply the constant multiple rule $$$\lim_{x \to \infty} c f{\left(x \right)} = c \lim_{x \to \infty} f{\left(x \right)}$$$ with $$$c=\frac{1}{6}$$$ and $$$f{\left(x \right)} = \frac{e^{x}}{x^{5}}$$$:
$${\color{red}{\lim_{x \to \infty} \frac{e^{x}}{6 x^{5}}}} = {\color{red}{\left(\frac{\lim_{x \to \infty} \frac{e^{x}}{x^{5}}}{6}\right)}}$$
Since we have an indeterminate form of type $$$\frac{\infty}{\infty}$$$, we can apply the l'Hopital's rule:
$$\frac{{\color{red}{\lim_{x \to \infty} \frac{e^{x}}{x^{5}}}}}{6} = \frac{{\color{red}{\lim_{x \to \infty} \frac{\frac{d}{dx}\left(e^{x}\right)}{\frac{d}{dx}\left(x^{5}\right)}}}}{6}$$
For steps, see derivative calculator.
$$\frac{{\color{red}{\lim_{x \to \infty} \frac{\frac{d}{dx}\left(e^{x}\right)}{\frac{d}{dx}\left(x^{5}\right)}}}}{6} = \frac{{\color{red}{\lim_{x \to \infty} \frac{e^{x}}{5 x^{4}}}}}{6}$$
Apply the constant multiple rule $$$\lim_{x \to \infty} c f{\left(x \right)} = c \lim_{x \to \infty} f{\left(x \right)}$$$ with $$$c=\frac{1}{5}$$$ and $$$f{\left(x \right)} = \frac{e^{x}}{x^{4}}$$$:
$$\frac{{\color{red}{\lim_{x \to \infty} \frac{e^{x}}{5 x^{4}}}}}{6} = \frac{{\color{red}{\left(\frac{\lim_{x \to \infty} \frac{e^{x}}{x^{4}}}{5}\right)}}}{6}$$
Since we have an indeterminate form of type $$$\frac{\infty}{\infty}$$$, we can apply the l'Hopital's rule:
$$\frac{{\color{red}{\lim_{x \to \infty} \frac{e^{x}}{x^{4}}}}}{30} = \frac{{\color{red}{\lim_{x \to \infty} \frac{\frac{d}{dx}\left(e^{x}\right)}{\frac{d}{dx}\left(x^{4}\right)}}}}{30}$$
For steps, see derivative calculator.
$$\frac{{\color{red}{\lim_{x \to \infty} \frac{\frac{d}{dx}\left(e^{x}\right)}{\frac{d}{dx}\left(x^{4}\right)}}}}{30} = \frac{{\color{red}{\lim_{x \to \infty} \frac{e^{x}}{4 x^{3}}}}}{30}$$
Apply the constant multiple rule $$$\lim_{x \to \infty} c f{\left(x \right)} = c \lim_{x \to \infty} f{\left(x \right)}$$$ with $$$c=\frac{1}{4}$$$ and $$$f{\left(x \right)} = \frac{e^{x}}{x^{3}}$$$:
$$\frac{{\color{red}{\lim_{x \to \infty} \frac{e^{x}}{4 x^{3}}}}}{30} = \frac{{\color{red}{\left(\frac{\lim_{x \to \infty} \frac{e^{x}}{x^{3}}}{4}\right)}}}{30}$$
Since we have an indeterminate form of type $$$\frac{\infty}{\infty}$$$, we can apply the l'Hopital's rule:
$$\frac{{\color{red}{\lim_{x \to \infty} \frac{e^{x}}{x^{3}}}}}{120} = \frac{{\color{red}{\lim_{x \to \infty} \frac{\frac{d}{dx}\left(e^{x}\right)}{\frac{d}{dx}\left(x^{3}\right)}}}}{120}$$
For steps, see derivative calculator.
$$\frac{{\color{red}{\lim_{x \to \infty} \frac{\frac{d}{dx}\left(e^{x}\right)}{\frac{d}{dx}\left(x^{3}\right)}}}}{120} = \frac{{\color{red}{\lim_{x \to \infty} \frac{e^{x}}{3 x^{2}}}}}{120}$$
Apply the constant multiple rule $$$\lim_{x \to \infty} c f{\left(x \right)} = c \lim_{x \to \infty} f{\left(x \right)}$$$ with $$$c=\frac{1}{3}$$$ and $$$f{\left(x \right)} = \frac{e^{x}}{x^{2}}$$$:
$$\frac{{\color{red}{\lim_{x \to \infty} \frac{e^{x}}{3 x^{2}}}}}{120} = \frac{{\color{red}{\left(\frac{\lim_{x \to \infty} \frac{e^{x}}{x^{2}}}{3}\right)}}}{120}$$
Since we have an indeterminate form of type $$$\frac{\infty}{\infty}$$$, we can apply the l'Hopital's rule:
$$\frac{{\color{red}{\lim_{x \to \infty} \frac{e^{x}}{x^{2}}}}}{360} = \frac{{\color{red}{\lim_{x \to \infty} \frac{\frac{d}{dx}\left(e^{x}\right)}{\frac{d}{dx}\left(x^{2}\right)}}}}{360}$$
For steps, see derivative calculator.
$$\frac{{\color{red}{\lim_{x \to \infty} \frac{\frac{d}{dx}\left(e^{x}\right)}{\frac{d}{dx}\left(x^{2}\right)}}}}{360} = \frac{{\color{red}{\lim_{x \to \infty} \frac{e^{x}}{2 x}}}}{360}$$
Apply the constant multiple rule $$$\lim_{x \to \infty} c f{\left(x \right)} = c \lim_{x \to \infty} f{\left(x \right)}$$$ with $$$c=\frac{1}{2}$$$ and $$$f{\left(x \right)} = \frac{e^{x}}{x}$$$:
$$\frac{{\color{red}{\lim_{x \to \infty} \frac{e^{x}}{2 x}}}}{360} = \frac{{\color{red}{\left(\frac{\lim_{x \to \infty} \frac{e^{x}}{x}}{2}\right)}}}{360}$$
Since we have an indeterminate form of type $$$\frac{\infty}{\infty}$$$, we can apply the l'Hopital's rule:
$$\frac{{\color{red}{\lim_{x \to \infty} \frac{e^{x}}{x}}}}{720} = \frac{{\color{red}{\lim_{x \to \infty} \frac{\frac{d}{dx}\left(e^{x}\right)}{\frac{d}{dx}\left(x\right)}}}}{720}$$
For steps, see derivative calculator.
$$\frac{{\color{red}{\lim_{x \to \infty} \frac{\frac{d}{dx}\left(e^{x}\right)}{\frac{d}{dx}\left(x\right)}}}}{720} = \frac{{\color{red}{\lim_{x \to \infty} e^{x}}}}{720}$$
The function grows without a bound:
$$\lim_{x \to \infty} e^{x} = \infty$$
Therefore,
$$\lim_{x \to \infty} \frac{e^{x}}{x^{6}} = \infty$$
Answer: $$$\lim_{x \to \infty} \frac{e^{x}}{x^{6}}=\infty$$$