Look at, for example, a function like which plots the velocity of an object against time.
Now, you will be aware that the displacement is given by the area under the graph (which can be found by integrating the function). Then, as time approaches infinity, we can evaluate the distance travelled:
So even as time becomes infinite, the velocity remains positive (though infinitely small) and the distance travelled remains finite.
We can confine the finite distance to any arbitrary value by using the velocity function:
It is also worth noting that there are other functions that also have finite areas (and v is positive for all t) for example: