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Question Number 175996 by sciencestudent last updated on 10/Sep/22

$$li\underset{x\to 0}{m}\frac{a^x-1}{x}=?$$

Answered by Ar Brandon last updated on 10/Sep/22

$$\underset{x\to 0}{\lim }\frac{a^x-1}{x}=\underset{x\to 0}{\lim }\frac{e^{x\ln a}-1}{x}$$$$=\underset{x\to 0}{\lim }\frac{(1+x\ln a)-1}{x}=\underset{x\to 0}{\lim }\frac{x\ln a}{x}=\ln a$$

Commented by sciencestudent last updated on 11/Sep/22

$$e^{xlnx}=1+xlnawhy?e^{xlna}=xlna$$

Commented by Ar Brandon last updated on 11/Sep/22

$$e^t\to 1+t+\frac{t^2}{2!}+\frac{t^3}{3!}+....+\frac{t^n}{n!}+...$$$$\mathrm{as}t\to 0$$

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