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Υπολογιστής Ορίων
Υπολογίστε όρια βήμα προς βήμα
Αυτός ο δωρεάν υπολογιστής θα προσπαθήσει να βρει το όριο (αμφίπλευρο ή μονόπλευρο, συμπεριλαμβανομένων του αριστερού και του δεξιού) της δοθείσας συνάρτησης στο δεδομένο σημείο (περιλαμβανομένου και του απείρου), με τα βήματα να παρουσιάζονται.
Χρησιμοποιούνται διάφορες τεχνικές για τον χειρισμό ορίων (συμπεριλαμβανομένων των απροσδιόριστων μορφών): νόμοι των ορίων, μετασχηματισμός και απλοποίηση, ο κανόνας του L'Hôpital, εξορθολογισμός του παρονομαστή, λήψη φυσικού λογαρίθμου κ.λπ.
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: $$$\lim_{x \to \infty} \frac{e^{x}}{x^{6}}=\infty$$$