Explanation

(c)
$\lim_{x \rightarrow 0}\frac{1 + \sin{x - \cos{x + \log{(1 - x)}}}}{x^{3}}$
$= \lim_{x \rightarrow 0}\frac{\begin{array}{r}
1 + \left( x - \frac{x^{3}}{3!} + \ldots \right) - \left( 1 - \frac{x^{2}}{2!} + \ldots \right) \\
- \left( x - \frac{x^{2}}{2} + \frac{x^{3}}{3} + ... \right)
\end{array}}{x^{3}}$
$= \lim_{x \rightarrow 0}\frac{- \frac{x^{3}}{2} + higher\ power\ of\ x}{x^{3}}$
$= - \frac{1}{2} + 0 = - \frac{1}{2}$