Explanation

(a)
${\lim_{x \rightarrow 0}\left( \frac{1 + 5x^{2}}{1 + 3x^{2}} \right)}^{1/x^{2}} = {\lim_{x \rightarrow 0}\left( 1 + \frac{2x^{2}}{1 + 3x^{2}} \right)}^{1/x^{2}}$
$= e^{\lim_{x \rightarrow 0}{\frac{1}{x^{2}}\left( \frac{2x^{2}}{1 + 3x^{2}} \right)}} = e^{2}$