Chapter 9.10, Problem 29ES

The limit of an indeterminate form as $x\to x_0$ can sometimes be found by expanding the functions involved in Taylor series about $x=x_0$ and taking the limit of the series term by term. Use this method to find the limits in these exercises.

$\begin{array}{l}\text{(a)}\underset{x\to 0}{\lim }\frac{\sin x}{x}\\ \left(\text{b}\right)\underset{x\to 0}{\lim }\frac{{\tan }^{-1}x-x}{x^3}\end{array}$