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  4. lim x → a (ax-xa/xx-aa)=-1 , then

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Q. $ \lim_ {{x \to a}} \frac {a^x-x^a}{x^x-a^a}=-1 $, then

Solution:

$\lim_{x\to a} \frac{a^{x} - x^{a}}{x^{x} -a^{a}} = \lim _{x\to a} \frac{a^{x} \, \log \,a - ax^{a-1}}{x^{x} \left(1 + \log \,x\right)}$ (L Hospital Rule) $= \frac{a^{a} \left(\log \, - 1\right)}{a^{a} \left(\log \, a+1\right)} =\frac{\log \,a -1}{\log \, a+1} = -1 $ (give) $ \Rightarrow \log \, a -1 = - \log \, a -1 $ $\Rightarrow 2\log \,a = 0 \Rightarrow a = 1$