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Question Number 107124 by Khalmohmmad last updated on 08/Aug/20

Answered by mathmax by abdo last updated on 08/Aug/20

$$\mathrm{let}\mathrm{f}\left(\mathrm{x}\right)=\frac{3^{\pi \mathrm{x}}-1}{\pi \mathrm{x}}\mathrm{we}\mathrm{have}{3}^{\pi \mathrm{x}}={\mathrm{e}}^{\pi \mathrm{xln}\left(3\right)}=1+\pi \mathrm{xln}\left(3\right)+\mathrm{o}\left({\mathrm{x}}^2\right)\Rightarrow$$$$\frac{{\mathrm{e}}^{\pi \mathrm{xln}3}-1}{\pi \mathrm{x}}=\ln \left(3\right)+\mathrm{o}\left(\mathrm{x}\right)\Rightarrow {\lim }_{\mathrm{x}\to 0}\mathrm{f}\left(\mathrm{x}\right)=\ln \left(3\right)$$

Answered by JDamian last updated on 08/Aug/20

$$3^{\pi x}=1+\frac{\pi \ln 3}{1!}x+\frac{{\left(\pi \ln 3\right)}^2}{2!}{x}^2+\cdot \cdot \cdot$$$$$$$$\underset{x\to 0}{\lim }\left(\frac{3^{\pi x}-1}{\pi x}\right)=\underset{x\to 0}{\lim }(\ln 3+\frac{{\left(\pi \ln 3\right)}^2}{2!\pi }x+\cdot \cdot \cdot )=$$$$=\ln 3$$

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