Explanation

(a)
$\lim_{x \rightarrow 0}\frac{(1 - cos\ 2x)\sin{5x}}{x^{2}\sin{3x}}$
$= \lim_{x \rightarrow 0}\frac{1 - cos\ 2x}{x^{2}}\lim_{x \rightarrow 0}\frac{\sin{5x}}{\sin{3x}}$
$= \lim_{x \rightarrow 0}\frac{2\sin^{2}x}{x^{2}}\lim_{x \rightarrow 0}\frac{\sin{5x}}{5x}.\frac{3x}{\sin{3x}}.\frac{5x}{3x}$
$= \lim_{x \rightarrow 0}2\left( \frac{\sin x}{x} \right)^{2}.\frac{5}{3} = \frac{10}{3}$