Explanation

(b)
$\lim_{x \rightarrow 1}\left( \log{ex} \right)^{1/\log x} = \lim_{x \rightarrow 1}\left\lbrack \log{e + \log x} \right\rbrack^{1/\log x}$
$= \lim_{x \rightarrow 1}\left\lbrack 1 + \log x \right\rbrack^{1/\log x}$
$= e^{\lim_{x \rightarrow 1}\ \frac{\log x}{\log x}} = e$