It might seem trivial but I keep asking myself whether there is a name for a cdf and its derivative that does not attain 1 in the limit, i.e. I‘m thinking of cases like
Let A, B, and C be the lifetime of some parts of a machine that are distributed iid on some support. The machine fails when 2 parts fail and the lifetime of the third part is then censored. Let G(x) be the observed distribution of lifetimes when A survives longest.
$$G(x)=\mathbb{P}[A>B,C \& B,C<x]$$
Clearly, when x goes to infinity, G(x) attains $1/3$. Is there a name for such distributions?
Your problem setup is wrong. One of the requirements on cdf is that is attains 1 as its limit, otherwise it is not a cdf :-)
In your case, you likely want to define $$ G(x) = \mathbb{P}\left[ \max\{B,C\} < x | A > \max\{B,C\}\right] $$ in which case $$ \lim_{x \to \infty} G(x) = 1, $$ as expected...