Explanation

(b)
$\lim_{x \rightarrow 0}\frac{\log{(1 + x^{3})}}{\sin^{3}x} = \lim_{x \rightarrow 0}\frac{\left( \frac{x^{3}}{1} - \frac{\left( x^{3} \right)^{2}}{2} + \frac{\left( x^{3} \right)^{3}}{3} - ...\infty \right)}{\left( x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - ...\infty \right)^{3}}$
$= \lim_{x \rightarrow 0}\frac{\left( 1 - \frac{x^{3}}{2} + \frac{\left( x^{3} \right)^{2}}{3} - ...\infty \right)}{\left( 1 - \frac{x^{2}}{3!} + \frac{x^{4}}{5!} - ...\infty \right)^{3}} = 1$