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Question Number 122350 by shaker last updated on 16/Nov/20

Answered by liberty last updated on 16/Nov/20

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

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