Explanation

(a)
$\lim_{x \rightarrow 0}\frac{a^{x} - b^{x}}{e^{x} - 1} = \lim_{x \rightarrow 0}\frac{a^{x} - b^{x}}{x}.\frac{x}{a^{x} - 1}$
$= \left\lbrack \lim_{x \rightarrow 0}\left( \frac{a^{x} - 1}{x} \right) - \lim_{x \rightarrow 0}\left( \frac{b^{x} - 1}{x} \right) \right\rbrack.\lim_{x \rightarrow 0}\frac{x}{e^{x} - 1}$
$= \left( \log_{e}{a - \log_{e}b} \right).\lim_{x \rightarrow 0}\frac{1}{\frac{e^{x} - 1}{x}}\$
$= \log_{e}\left( \frac{a}{b} \right)$