Negligible function

Question

I'm studying about complexity and reaching negligible. Can anyone tell me if $f(x)$ is a negligible function and $p(x) \in \mathbb{R}$ then $p(x) . f(x)$ is a negligible function?

Answer 1

Let $f'(x) = f(x) \cdot p(x)$

1. f(x) is negligible means that it is smaller than the inverse of any polynomial, for all sufficiently large n. In your case, given any polynomial $q(x)$, it is smaller than $1/p(x)q(x)$

2. By using the limit definition of negligibility.

   $f(x)$ is negligible than for every polynomial $q(x)$ we have;

   $$\lim_{x \rightarrow \infty} q(x) f(x) =0$$

   We need to show that for every $q(x)$, $f'(x)$ is negligible;

   $$\lim_{n \rightarrow \infty} q(x) f'(x) = \lim_{x \rightarrow \infty} q(x) [p(x) f(x)] =0$$

   This is true; since $f(x)$ is negligible implies for any polynomial;

   $$\lim_{n \rightarrow \infty} [q(x) p(x)] f(x) =0,$$ where $q(x) p(x)$ is a polynomial.