How do you prove that if f(x) and g(x) are negligible, then f(x)/g(x) is not? I can use an example: the case where we have $x$ such that $f(x)=g(x)$. The quotient is $1$, a non-negligible function. However, we can't conclude that all functions $f(x)/g(x)$ are also negligible.

How can I formally prove this?
This is false, take $f(x) = 2^{-2x}$ and $g(x) = 2^{-x}$.