Let $R$ be a commutative ring with $1 \neq 0$ and let $P \subset R$ be a prime ideal. Apparently we have $$\varinjlim\limits_{f \in R \setminus P} R_f \cong R_P$$ where $R_f$ the the localization of $R$ at the set $\{1,f,f^2,\ldots\}$. Why is this the case?
edit 1: I'd like to use the construction of the direct limit and avoid the universal property.
edit 2: In case it's unclear, $R_P$ is the localization at the prime $P$ i.e. $R_P = (R \setminus P)^{-1}R$.