According to scientists, comets are leftover debris from the formation of the solar system 4.6 billion years ago, when the solar nebula condensed, and are thought to consist mostly of ice, dust and rock. This means that these leftover debris are as old as our solar system.
Short period comets are comets with orbital periods less than 200 years. They are believed to originate in a region of our solar system known as the Kuiper Belt, which lies just beyond the orbit of Neptune.
The leftover debris residing in the Kuiper Belt are thrown (or nudged) from these outer reaches of the Solar System towards the Sun by gravitational perturbations from the outer planets (i.e., Jupiter, Saturn, Uranus, Neptune) or as a result of collisions between objects within this region. Unless the leftover debris are nudged, they remain in suspended animation in the Kuiper Belt, being preserved there in cold storage for billions of years. Only those leftover debris that were nudged from their postions in the Kuiper Belt are the ones which eventually become short period comets.
Each time a comet, which is akin to a dirty snowball, passes near the sun it loses tons of material to vaporization. Thus, the number of orbits such a comet can make before being reduced to a swarm of gravel is limited. To give you an idea just how long these short period comets can exist, let me point out that the projected life span of one average sized short-period comet, that of Halley's comet, is only 40,000 years. Some scientists, including David C. Jewitt, estimates the sublimation and dynamical lifetimes of short period comets to be between a minimum of only 10,000 years to a maximum of 450,000 years.
Now consider this. Comets are supposed to have been formed 4.6 billion years ago. During this looong period of time, any leftover debris residing in the Kuiper Belt which could be nudged towards the Sun by gravitational perturbations from the outer planets (i.e., Jupiter, Saturn, Uranus, Neptune) or as a result of collisions between objects within this region should have been nudged by now. And, once these leftover debris are nudged to become short period comets, they are expected to last only around 10,000 - 450,000 years. In other words, by now, we should not be seeing any of these short period comets anymore, nada. And yet we still do.