L1 L2 Points: Influence & Model Dependence

[wolram]

Can some one tell me how big an area of influence the lagrange L1, L2
etc points have ? or are they model depandant ? Thanks.

 

[wolram]

http://www.physics.montana.edu/faculty/cornish/lagrange.html

In this diagram the area covered by the L4, L5 points look huge, but i guess a body would have to enter it slowly and near the center to be captured, but what would its maximum orbit be?
I can not understand how the L3 point is sustained, surly if the speed of G=C then any planetary influence would be (washed) away by the refresh rate of the speed of G ?

 

[D H]

The attraction is quite weak. 2002 AA29 sails right through the Earth-Sun L4 and L5 points in its bizarre horseshoe orbit.