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Question Number 30511 by abdo imad last updated on 22/Feb/18

find  lim_(x→+∞)   e^x  [(1/x)].

$${find}\:\:{lim}_{{x}\rightarrow+\infty} \:\:{e}^{{x}} \:\left[\frac{\mathrm{1}}{{x}}\right]. \\ $$

Commented by abdo imad last updated on 24/Feb/18

for x>1 ⇒ 0<(1/x)<1 ⇒ [(1/x)]=0 ⇒ ∀x>1 e^x  [(1/x)]=0⇒  lim_(x→+∞) e^x [(1/x)]=0 so the function [..] defeat e^x  .

$${for}\:{x}>\mathrm{1}\:\Rightarrow\:\mathrm{0}<\frac{\mathrm{1}}{{x}}<\mathrm{1}\:\Rightarrow\:\left[\frac{\mathrm{1}}{{x}}\right]=\mathrm{0}\:\Rightarrow\:\forall{x}>\mathrm{1}\:{e}^{{x}} \:\left[\frac{\mathrm{1}}{{x}}\right]=\mathrm{0}\Rightarrow \\ $$$${lim}_{{x}\rightarrow+\infty} {e}^{{x}} \left[\frac{\mathrm{1}}{{x}}\right]=\mathrm{0}\:{so}\:{the}\:{function}\:\left[..\right]\:{defeat}\:{e}^{{x}} \:. \\ $$

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