5. For the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$, the center, right focus, right vertex, and the intersection of the right directrix with the $x$-axis are denoted as $O, F, G, H$ respectively. Then the maximum value of $\frac{|F G|}{|O H|}$ is ( ).
(A) $\frac{1}{2}$
(B) $\frac{1}{3}$
(C) $\frac{1}{4}$
(D) Uncertain
5.C.
$$
\frac{F G}{O H}=\frac{a-c}{\frac{a^{2}}{c}}=\frac{a c-c^{2}}{a^{2}}=e-e^{2}=e(1-e) \leqslant \frac{1}{4},
$$

The equality holds if and only if $e=\frac{1}{2}$.