Answer
True
Work Step by Step
The only way to make $\lim\limits_{x \to 0} x*f(x) \ne 0$ true is if $\lim\limits_{x \to 0} f(x) = \infty$.
The same can be said to make $\lim\limits_{x \to +\infty} \frac{x}{f(x)} \ne 0$ true.
Then we would have $0*\infty$ and $\frac{\infty}{\infty}$, and the limits would be indeterminate, but since $f(x)$ is confined between finite numbers $-M$ and $M$, this would be impossible. Therefore, the statement is true.
