Explanation

(a)
$\lim_{x \rightarrow 0}\frac{a^{x} + a^{- x} - 2}{x^{2}} = \lim_{x \rightarrow 0}\frac{a^{x}\log a - a^{- x}\log a}{2x}$
[by L’ Hospital’s rule]
$= \lim_{x \rightarrow 0}\frac{a^{x}{{(log}{a)}}^{2} + a^{- x}{{(log}{a)}}^{2}}{2}$
$= \left( \log a \right)^{2}$ [by L’ Hospital’s rule]