|
|
|
|
|
|
|
|
|
|
|
|
Kalkulator Limit
Hitung limit langkah demi langkah
Kalkulator gratis ini akan mencoba mencari limit (dua sisi maupun satu sisi, termasuk kiri dan kanan) dari fungsi yang diberikan pada titik yang ditentukan (termasuk tak hingga), beserta langkah-langkahnya.
Berbagai teknik digunakan untuk menangani limit (termasuk bentuk tak tentu): aturan-aturan limit, penulisan ulang dan penyederhanaan, aturan L'Hôpital, merasionalisasi penyebut, mengambil logaritma natural, dan sebagainya.
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: $$$\lim_{x \to \infty} \frac{e^{x}}{x^{6}}=\infty$$$