If you know that |f(x)| <= C and |g(x)| <= C, then it is easy to find an upper bound on |f(x) * g(x)|.

On the other hand, f(x) / g(x) can be very large even if |g(x)| <= 1. In fact, in this case |f(x) / g(x)| can be made arbitrarily large by keeping f(x) constant and changing g(x) a little (by no more than 1).