Chapter 1.6, Problem 38ES

Determine whether the statement is true or false. Explain your answer.

Suppose that $M$ is a positive number and that for all real numbers $x$ , a function $f$ satisfies $-M\le f\left(x\right)\le M$ Then $\underset{x\to 0}{\lim }xf\left(x\right)=0$ and $\underset{x\to +\infty }{\lim }\frac{f\left(x\right)}{x}=0$