6 A particle \(P\) of mass \(m\) moves along the positive \(x\)-axis. When its displacement from the origin \(O\) is \(x\) its velocity is \(v\), where \(v \geqslant 0\). It is subject to two forces: a constant force \(T\) in the positive \(x\) direction, and a resistive force which is proportional to \(v ^ { 2 }\).

1. Show that \(v ^ { 2 } = \frac { 1 } { k } \left( T - A \mathrm { e } ^ { - \frac { 2 k x } { m } } \right)\) where \(A\) and \(k\) are constants.
   \(P\) starts from rest at \(O\).
2. Find an expression for the work done against the resistance to motion as \(P\) moves from \(O\) to the point where \(x = 1\).
3. Find an expression for the limiting value of the velocity of \(P\) as \(x\) increases.