Question Number 211812 by mokys last updated on 21/Sep/24
$${prove}\:\underset{{x}\rightarrow\infty} {{lim}}\:\left(\:\mathrm{1}\:+\:\frac{\mathrm{5}}{{x}}\:\right)^{\frac{\mathrm{1}}{{x}}} −\:\mathrm{1}\:=\:\mathrm{5}\: \\ $$
Commented by mr W last updated on 22/Sep/24
$${wrong}! \\ $$$${the}\:{result}\:{should}\:{be}\:\mathrm{0}. \\ $$
Answered by mathmax last updated on 23/Sep/24
$$\left(\mathrm{1}+\frac{\mathrm{5}}{{x}}\right)^{\frac{\mathrm{1}}{{x}}} \sim\mathrm{1}+\frac{\mathrm{5}}{{x}^{\mathrm{2}} }\:\Rightarrow\left(\mathrm{1}+\frac{\mathrm{5}}{{x}}\right)^{\frac{\mathrm{1}}{{x}}} −\mathrm{1}\sim\frac{\mathrm{5}}{{x}^{\mathrm{2}} }\left({x}\rightarrow+\infty\right)\:\Rightarrow \\ $$$${lim}_{{x}\rightarrow+\infty} \left(\mathrm{1}+\frac{\mathrm{5}}{{x}}\right)^{\frac{\mathrm{1}}{{x}}} −\mathrm{1}\:=\mathrm{0} \\ $$