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lim-e-x-60-




Question Number 143012 by Study last updated on 08/Jun/21
lim_(α→∞) (e^x /α^(60!) )=?
$${li}\underset{\alpha\rightarrow\infty} {{m}}\frac{{e}^{{x}} }{\alpha^{\mathrm{60}!} }=? \\ $$
Commented by Study last updated on 08/Jun/21
is the e^x  number?
$${is}\:{the}\:{e}^{{x}} \:{number}? \\ $$
Answered by mathmax by abdo last updated on 09/Jun/21
for  x fixed lim_(α→∞)  (e^x /α^(60!) )=0  perhaps the Q is lim_(x→+∞)  (e^x /x^(60!) )
$$\mathrm{for}\:\:\mathrm{x}\:\mathrm{fixed}\:\mathrm{lim}_{\alpha\rightarrow\infty} \:\frac{\mathrm{e}^{\mathrm{x}} }{\alpha^{\mathrm{60}!} }=\mathrm{0} \\ $$$$\mathrm{perhaps}\:\mathrm{the}\:\mathrm{Q}\:\mathrm{is}\:\mathrm{lim}_{\mathrm{x}\rightarrow+\infty} \:\frac{\mathrm{e}^{\mathrm{x}} }{\mathrm{x}^{\mathrm{60}!} } \\ $$
Commented by Study last updated on 09/Jun/21
what is the practice?
$${what}\:{is}\:{the}\:{practice}? \\ $$
Commented by mathmax by abdo last updated on 09/Jun/21
f(x)=(e^x /x^(60!) ) ⇒log(f(x))=x−60!logx =x(1−60!((logx)/x)) ⇒  lim_(x→+∞) log(f(x))=+∞ ⇒lim_(x→+∞) f(x)=+∞
$$\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{e}^{\mathrm{x}} }{\mathrm{x}^{\mathrm{60}!} }\:\Rightarrow\mathrm{log}\left(\mathrm{f}\left(\mathrm{x}\right)\right)=\mathrm{x}−\mathrm{60}!\mathrm{logx}\:=\mathrm{x}\left(\mathrm{1}−\mathrm{60}!\frac{\mathrm{logx}}{\mathrm{x}}\right)\:\Rightarrow \\ $$$$\mathrm{lim}_{\mathrm{x}\rightarrow+\infty} \mathrm{log}\left(\mathrm{f}\left(\mathrm{x}\right)\right)=+\infty\:\Rightarrow\mathrm{lim}_{\mathrm{x}\rightarrow+\infty} \mathrm{f}\left(\mathrm{x}\right)=+\infty \\ $$

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