Explanation

(a)
Given, $\ \lim_{x \rightarrow a}{\frac{a^{x} - x^{a}}{x^{x} - a^{a}} = - 1}$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \lim_{x \rightarrow a}\frac{a^{x}\log_{e}a - ax^{a - 1}}{x^{x}(1 + \log_{e}x) - 0} = - 1$ [by L’ Hospital’s rule]
$\Rightarrow \ \ \ \ \ \ \ \ \ {\frac{a^{a}\log_{e}a - a^{a}}{a^{a}(1 + \log_{e}a)} = - 1}$
$\Rightarrow \ \ \ \ \ \ \ \ \ 2\log_{e}{a = 0}\ \ \ \ \ \ \Rightarrow \ \ \ \ \ \ a = 1$