My old Elmark ceiling fan only rotate at low RPM eventhrough set at speed no 3 ? This condition is very obvious when I compare it again the newly install ceiling fan ,the speed is almost 60 percent slower compare to new fan . Is this low speed is due to capacitor problem ? Any method to improve the rotation speed ?

There is no way to know without testing it with a meter, but if the capacitor was bad it probably would not run at all. More than likely you have a bad switch. You can check it with an ohm meter. It switches power to one of three outputs. If it's bad it will only go to one output terminal.