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Question Number 206808 by mustafazaheen last updated on 26/Apr/24

$$\underset{x\to 0}{\lim }\frac{{10}^{\mathrm{x}}-1}{{\mathrm{x}}^{10}}$$

Answered by mr W last updated on 26/Apr/24

$${10}^x=e^{x\ln 10}=1+\frac{x\ln 10}{1!}+\frac{{\left(x\ln 10\right)}^2}{2!}+...$$$$\underset{x\to 0}{\lim }\frac{{10}^x-1}{x^{10}}$$$$=\underset{x\to 0}{\lim }\frac{\frac{x\ln 10}{1!}+\frac{{\left(x\ln 10\right)}^2}{2!}+...}{x^{10}}$$$$=\underset{x\to 0}{\lim }\frac{\ln 10}{x^{9}1!}+\frac{{\left(\ln 10\right)}^2}{x^{8}2!}+...$$$$=\mathrm{\infty }$$

Answered by lepuissantcedricjunior last updated on 26/Apr/24

$$\underset{x\to 0}{\lim }\frac{{10}^{\mathit{x}}-1}{{\mathit{x}}^{10}}=\frac{0}{0}=\mathbf{Fi}$$$$\underset{x\to 0}{\lim }\frac{{10}^{\mathbf{x}}-1}{{\mathbf{x}}^{10}}=\underset{x\to 0}{\lim }\frac{\mathbf{ln}10\times {10}^{\mathbf{x}}}{10{\mathbf{x}}^9}=\frac{\mathbf{ln}10}{0^{\mp }}=\mp \mathrm{\infty }$$

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