1. The problem asks to evaluate the limit $ \lim_{x \to 3^+} [f(x)g(x)] $. 2. To solve limits involving products, use the property: $$\lim_{x \to a} [f(x)g(x)] = \lim_{x \to a} f(x) \times \lim_{x \to a} g(x)$$ if both limits exist. 3. From the given answers, $ \lim_{x \to 3^+} [f(x)g(x)] = \text{DNE} $ means the limit does not exist. 4. This implies either $ \lim_{x \to 3^+} f(x) $ or $ \lim_{x \to 3^+} g(x) $ or both do not exist or the product oscillates without limit. 5. Since the problem states DNE, the final answer is: $$\boxed{\text{DNE}}$$