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Question Number 88610 by M±th+et£s last updated on 11/Apr/20

is 1×1×1×..........=1^∞    or   1×1×1×1×1×..........=1

$${is}\:\mathrm{1}×\mathrm{1}×\mathrm{1}×..........=\mathrm{1}^{\infty} \: \\ $$$${or} \\ $$$$\:\mathrm{1}×\mathrm{1}×\mathrm{1}×\mathrm{1}×\mathrm{1}×..........=\mathrm{1} \\ $$$$ \\ $$$$ \\ $$

Commented by mahdi last updated on 11/Apr/20

for example:  A)lim_(x→∞) (1+(1/x))^x =1^∞ =e≠1  B)lim_(x→∞) (1^x )=1^∞ =1=1  in A: 1×1×...=1^∞ ≠1    B:1×1×...=1^∞ =1  Question is important.

$$\mathrm{for}\:\mathrm{example}: \\ $$$$\left.\mathrm{A}\right)\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{x}}\right)^{\mathrm{x}} =\mathrm{1}^{\infty} =\mathrm{e}\neq\mathrm{1} \\ $$$$\left.\mathrm{B}\right)\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\mathrm{1}^{\mathrm{x}} \right)=\mathrm{1}^{\infty} =\mathrm{1}=\mathrm{1} \\ $$$$\mathrm{in}\:\mathrm{A}:\:\mathrm{1}×\mathrm{1}×...=\mathrm{1}^{\infty} \neq\mathrm{1}\:\:\:\:\mathrm{B}:\mathrm{1}×\mathrm{1}×...=\mathrm{1}^{\infty} =\mathrm{1} \\ $$$$\mathrm{Question}\:\mathrm{is}\:\mathrm{important}. \\ $$

Commented by M±th+et£s last updated on 11/Apr/20

and there is more examples like  lim_(x→∞) (1+(1/x^2 ))^x =1  lim_(x→∞) (1+(1/(log(x))))=∞  lim_(x→∞) (1−(1/(log(x))))^x =0    so i think that its not wrong if we say that1×1×1×1×.....=1^∞ =1  but not in every time 1×1×1×1.....=1^∞

$${and}\:{there}\:{is}\:{more}\:{examples}\:{like} \\ $$$$\underset{{x}\rightarrow\infty} {{lim}}\left(\mathrm{1}+\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right)^{{x}} =\mathrm{1} \\ $$$$\underset{{x}\rightarrow\infty} {{lim}}\left(\mathrm{1}+\frac{\mathrm{1}}{{log}\left({x}\right)}\right)=\infty \\ $$$$\underset{{x}\rightarrow\infty} {{lim}}\left(\mathrm{1}−\frac{\mathrm{1}}{{log}\left({x}\right)}\right)^{{x}} =\mathrm{0} \\ $$$$ \\ $$$${so}\:{i}\:{think}\:{that}\:{its}\:{not}\:{wrong}\:{if}\:{we}\:{say}\:{that}\mathrm{1}×\mathrm{1}×\mathrm{1}×\mathrm{1}×.....=\mathrm{1}^{\infty} =\mathrm{1} \\ $$$${but}\:{not}\:{in}\:{every}\:{time}\:\mathrm{1}×\mathrm{1}×\mathrm{1}×\mathrm{1}.....=\mathrm{1}^{\infty} \\ $$$$ \\ $$$$ \\ $$

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