Let's assume a thin lens and an object O in focus at a distance of dO from the lens. Further, let's assume there's a large out-of-focus background object B at a distance of dB. Now, we know intuitively, and can easily derive from the thin lens formula, that as dB tends to infinity, the image of B on the projection surface asymptotically approaches some finite amount of "blurriness" -- the focused image of B is formed at some finite distance from the projection surface, and by straightforward geometry, points of B are projected as blur circles whose diameter is proportional to that distance.
However, as dB grows, the "amount" of background visible in the image grows proportionally to it. When striving to achieve a minimally distracting background, we often seek to minimize the amount of background visible while maximizing the blur. My question is: Is there a rule of thumb for figuring out an "optimal" dB as a function of dO that does just that? Obviously the focal length and aperture size of the lens matter greatly here, but let's assume that these are fixed andwe can only change dB.