lim-x-a-x-a-a-x-a-x-a-a-a-gt-0-

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

\lim_{x \to a} \frac{x^{a} - a^{x}}{a^{x} - a^{a}} =; a > 0

Answered by liberty last updated on 24/Dec/20

\lim_{x \to a} \frac{\frac{d}{d x} (x^{a} - a^{x})}{\frac{d}{d x} (a^{x} - a^{a})} = \lim_{x \to a} \frac{{a x}^{a - 1} - \ln a . (a^{x})}{\ln a . (a^{x})}

= \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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