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

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Question Number 208312 by messele last updated on 11/Jun/24
lim_(x→0 ((a^x −1)/x) = log a)
limx→0ax−1x=loga
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
letf(x)=ax=exlna⇒f(0)=1andlimx→0ax−1x=f′(0)orf′(x)=lna.ax⇒f′(0)=lna⇒limx→0ax−1x=lna

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