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Question Number 144772 by phally last updated on 29/Jun/21

Answered by liberty last updated on 29/Jun/21

$$\underset{x\to 1}{\lim }\frac{\frac{\mathrm{d}}{\mathrm{dx}}({\mathrm{x}}^{\mathrm{x}}-1)}{\frac{\mathrm{d}}{\mathrm{dx}}(\ln (1-\sqrt{{\mathrm{x}}^2-1})}$$$$\mathrm{consider}\frac{\mathrm{d}}{\mathrm{dx}}({\mathrm{x}}^{\mathrm{x}}-1)={\mathrm{x}}^{\mathrm{x}}(\ln \mathrm{x}+1)$$$$\mathrm{and}\frac{\mathrm{d}}{\mathrm{dx}}\left(\ln (1-\sqrt{{\mathrm{x}}^2-1})\right)=\frac{-\frac{\mathrm{x}}{\sqrt{{\mathrm{x}}^2-1}}}{1-\sqrt{{\mathrm{x}}^2-1}}$$$$\mathrm{so}\underset{x\to 1}{\lim }\frac{{\mathrm{x}}^{\mathrm{x}}(\ln \mathrm{x}-1)(1-\sqrt{{\mathrm{x}}^2-1})\left(\sqrt{{\mathrm{x}}^2-1}\right)}{-\mathrm{x}}$$$$=0$$

Commented by phally last updated on 29/Jun/21

$$\mathrm{thank}\mathrm{brother}$$

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