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lim-x-0-a-x-1-x-log-a-




Question Number 208312 by messele last updated on 11/Jun/24
lim_(x→0 ((a^x −1)/x) = log a)
$${lim}_{{x}\rightarrow\mathrm{0}\:\frac{{a}^{{x}} −\mathrm{1}}{{x}}\:=\:{log}\:{a}} \\ $$
Answered by mathzup last updated on 11/Jun/24
let f(x)=a^x  =e^(xlna)  ⇒f(0)=1 and  lim_(x→0) ((a^x −1)/x)=f^′ (0) or f^′ (x)=lna.a^x  ⇒  f^′ (0)=lna ⇒lim_(x→0) ((a^x −1)/x)=lna
$${let}\:{f}\left({x}\right)={a}^{{x}} \:={e}^{{xlna}} \:\Rightarrow{f}\left(\mathrm{0}\right)=\mathrm{1}\:{and} \\ $$$${lim}_{{x}\rightarrow\mathrm{0}} \frac{{a}^{{x}} −\mathrm{1}}{{x}}={f}^{'} \left(\mathrm{0}\right)\:{or}\:{f}^{'} \left({x}\right)={lna}.{a}^{{x}} \:\Rightarrow \\ $$$${f}^{'} \left(\mathrm{0}\right)={lna}\:\Rightarrow{lim}_{{x}\rightarrow\mathrm{0}} \frac{{a}^{{x}} −\mathrm{1}}{{x}}={lna} \\ $$

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