6.68. Using Taylor and Maclaurin formulas, compute the limits

a) $\lim _{x \rightarrow 0} \frac{\sin x-x}{x^{3}}$

b) $\lim _{x \rightarrow 0} \frac{\ln (1+x)-x+x^{2} / 2}{x^{3}}$.
Solution.

a) $\lim _{x \rightarrow 0} \frac{\left(x-x^{3} / 6+o\left(x^{4}\right)\right)-x}{x^{3}}=\lim _{x \rightarrow 0}\left(\frac{1}{6}+\frac{o\left(x^{4}\right)}{x^{3}}\right)=\frac{1}{6}$;

$$
\begin{gathered}
\lim _{x \rightarrow 0} \frac{\ln (1+x)-x+x^{2} / 2}{x^{3}}= \\
=\lim _{x \rightarrow 0} \frac{\left(x-x^{2} / 2+x^{3} / 3+o\left(x^{3}\right)\right)-x+x^{2} / 2}{x^{3}}= \\
=\lim _{x \rightarrow 0}\left(\frac{1}{3}+\frac{o\left(x^{3}\right)}{x^{3}}\right)=\frac{1}{3} .
\end{gathered}
$$

## FINDING INTERVALS OF MONOTONICITY AND EXTREMA OF THE FUNCTION $f(x)$