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lim-x-0-log-e-x-1-x-please-help-




Question Number 150759 by nimnim last updated on 15/Aug/21
  lim_(x→0) ((log(e+x)−1)/x)=?  please help..
$$\:\:\underset{{x}\rightarrow\mathrm{0}} {{lim}}\frac{{log}\left({e}+{x}\right)−\mathrm{1}}{{x}}=? \\ $$$${please}\:{help}.. \\ $$
Answered by Olaf_Thorendsen last updated on 15/Aug/21
((ln(e+x)−1)/x) = ((lne+ln(1+(x/e))−1)/x)  = ((ln(1+(x/e)))/x) ∼_0  ((x/e)/x) = (1/e)
$$\frac{\mathrm{ln}\left({e}+{x}\right)−\mathrm{1}}{{x}}\:=\:\frac{\mathrm{ln}{e}+\mathrm{ln}\left(\mathrm{1}+\frac{{x}}{{e}}\right)−\mathrm{1}}{{x}} \\ $$$$=\:\frac{\mathrm{ln}\left(\mathrm{1}+\frac{{x}}{{e}}\right)}{{x}}\:\underset{\mathrm{0}} {\sim}\:\frac{\frac{{x}}{{e}}}{{x}}\:=\:\frac{\mathrm{1}}{{e}} \\ $$
Commented by nimnim last updated on 15/Aug/21
Thank you so much.,Sir.
$${Thank}\:{you}\:{so}\:{much}.,{Sir}. \\ $$

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