How far ahead or behind can astronomers calculate planets positions? What are limits of these calculations?
Depending on the required accuracy, the positions of the planets can be calculated for hundreds or thousands of years ahead and behind.
The VSOP87 data can be used to calculate the positions of planets for any date between 2000BC and 6000AD. This data is good enough for most purposes, including almanacs and ephemerides.
The NASA DE430 data can be used to calculate the positions of the Sun, Moon and the planets for dates between 1550 and 2650 to a high degree of accuracy. This data is used to get spacecraft and space probes to their correct positions.