1-F(x) as F(x) goes to 1

Question

I stumbled into the following statement and I am not sure how to prove it. The statement is :

Given a distribution function $F(x)$ we can represent the survival function $1-F(x)$ as $-logF(x)$ for $F(x)\rightarrow 1$.

More explicitly $1-F(x)\sim{-logF(x)}$ as $F(x)\rightarrow 1.$

Any references would be appreciated.Thank you.

Answer 1

We cannot. What you are referring to is probably the fact that $\frac{-\log u}{1-u}\to1$ when $u\to1$.