Yes. If a foul ball is caught by a defensive player before it hits the ground the batter is out.
There is no limit to the number of fouls in the MLB.The only rules about foul balls is if a batter attempts to bunt with two strikes and the ball is bunted foul, the batter is called out. But there is no limit to the number of foul balls a batter can hit when swinging at the pitch.This is true. However, the first 2 foul balls that are hit and not caught are considered strikes. After that, there is no limit to the amount of foul balls that can be hit, as long as they're not caught.
A Baseball player hitting foul after foul is done more so to stay alive (not strike out) then to work the pitcher ... You have to keep in mind that the batter him self is using up energy to hit the ball foul ... The batter would rather get a hit then repetitively hitting the ball foul ...
If you are on base, and the batter hits a foul ball that is caught, and there is less than 2 outs, you may attempt to run to the next base after the ball is caught.
There is no limit; however, if a foul ball is caught by the other team, then the batter is out.
If the baseball hit the ground in foul territory, the ball is a foul ball. If the baseball hit the ground in fair territory, and the batter/runner is still in the batter's box when the baseball hits him, it is a foul ball. If the baseball hit the ground in fair territory and the batter/runner is out of the batter's box when the baseball hits him, the batter/runner is out and the ball is dead.