I'm looking for examples of distributions for which $\frac{1 - F(t + \Delta t)}{1 - F(t)} \to 0$ as $t \to + \infty$, $\forall \Delta t >0$. Here $F$ is the cdf.
It looks like exponential doesn't work (memoryless). Gumbel distribution doesn't work either. Are there famous distributions for which this would work?