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Question Number 126801 by john_santu last updated on 24/Dec/20

$$\underset{x\to a}{\lim }\frac{x^a-a^x}{a^x-a^a}=;a>0$$ $$$$

Answered by liberty last updated on 24/Dec/20

$$\underset{x\to a}{\lim }\frac{\frac{d}{dx}(x^a-a^x)}{\frac{d}{dx}(a^x-a^a)}=\underset{x\to a}{\lim }\frac{{ax}^{a-1}-\ln a.\left(a^x\right)}{\ln a.\left(a^x\right)}$$ $$=\frac{a.a^{a-1}-a^a.\ln a}{a^a.\ln a}=\frac{a^a(1-\ln a)}{a^a.\ln a}=\frac{1-\ln a}{\ln a}$$

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