I believe it's nearly universal that the "L" is the last one to be accepted. Some never do, hence 4-pointers otherwise known as Amyrauldians after a guy named Amyrault.
It was basically the same for me. The strange thing was that once I saw what limited atonement was really about, that is that it is not so much who Christ did not die for as it is about who Christ DID die for and the atonements real purpose, it became the easiest one to accept. The P became the last one for me to accept fully, oddly enough, but that's settled now.
The L was not an obstacle for me accepting the other points, so I saw early on that I was going to be comfortable with T, U, I, and 1/2 of P (accepting preservation, but not preserverence).
What changed my mind finally to accept the L was realizing that the atonement was the SURETY of my salvation - it was the GUARANTEE of forgiveness to the elect. A general atonement could not provide that.
Also, it is usually the last one to be accepted because many of those scriptures which seem to have a universal application are associated with the atonement. When it becomes clear that "world" does not mean "every person that ever lived" and that "all" does not necessarily mean "each and every person" then it becomes scripturally tenable that the atonement has a limited scope.
"I lay down my life for THE SHEEP"
"by whome WE have now received the atonement"
"
Romans 5:6 For when WE were yet without strength, in due time Christ died for the ungodly.
Romans 5:8 But God commendeth his love toward US, in that, while WE were yet sinners, Christ died for US.
Romans 14:15 But if THY BROTHER be grieved with [thy] meat, now walkest thou not charitably. Destroy not him with thy meat, for whom Christ died.
1 Cor 8:11 And through thy knowledge shall the WEAK BROTHER perish, for whom Christ died?
1 Cor 15:3 For I delivered unto you first of all that which I also received, how that Christ died for OUR sins according to the scriptures;
