Final answer:
If lim x→7 f(x) = 0 and lim x→7 g(x) = 0, then lim x→7 f(x) g(x) does not exist is true.
Step-by-step explanation:
The statement given in the question is True.
If both lim x→7 f(x) and lim x→7 g(x) are equal to 0, it does not guarantee that lim x→7 f(x) g(x) exists. To determine if the limit exists, we need to consider the behavior of the product of two functions near the point x = 7. It is possible for the product of two functions to have a finite limit, approach infinity, or not have a limit at all.
For example, if f(x) = 1/x and g(x) = x - 7, both lim x→7 f(x) = 0 and lim x→7 g(x) = 0, but lim x→7 f(x) g(x) does not exist.