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

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