Answer
$$f(x)=\begin{cases}1, & x>0\\ -1 & x \le 0\end{cases}$$
Work Step by Step
Consider the following function:$$f(x)=\begin{cases}1, & x>0\\ -1 & x \le 0\end{cases}.$$It is clear that $\lim_{x \to 0}f(x)$ does not exist. But, $|f(x)|$ is a constant function, $|f(x)|=1$, and has limit at any point, so we have$$\lim_{x \to 0}|f(x)|=1.$$ Thus, we have found a function
showing that the converse of Exercise 112(b) is not true.
