Answer
Please see below.
Work Step by Step
We want to prove that $$\lim_{x \to c}|f(x)|=0$$ by using $\epsilon - \delta$ definition; that is, we must show that for each $\epsilon >0$, there exists a $\delta >0$ such that $||f(x)|-0|< \epsilon$ whenever $|x-c|< \delta$.
By the assumption, $\lim_{x \to c} f(x) =0$; that is, for any $\epsilon > 0$, there exists a $\delta >0$ such that$$|x-c|< \delta \quad \Rightarrow \quad |f(x)-0|=|f(x)|< \epsilon .$$So we can conclude that the given statement is true.
