|
|
|
|
|
|
|
|
|
|
|
|
Limietcalculator
Bereken limieten stap voor stap
Deze gratis rekenmachine probeert de limiet (tweezijdig of eenzijdig, links of rechts) van de gegeven functie in het gegeven punt (inclusief oneindig) te vinden, waarbij de stappen worden getoond.
Er worden verschillende technieken gebruikt om limieten te behandelen (waaronder onbepaalde vormen): limietwetten, herschrijven en vereenvoudigen, de regel van L'Hôpital, het rationaliseren van de noemer, het nemen van de natuurlijke logaritme, enz.
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: $$$\lim_{x \to \infty} \frac{e^{x}}{x^{6}}=\infty$$$