The term'limited atonement' is not actually a helpful term. Every theological position that advocates that Jesus Christ provides an atonement applies some sort of limitation to that atonement with the exception of universalism.
For whom did Jesus die
The question really boils down to who did Jesus die for, did he die to achieve the actual salvation of the elect or did he deny to provide a potential salvation to those who are willing to receive it?
Joseph was instructed to call him Jesus because he would "save his people from their sins" matt 1:21
In John 10:11 Jesus Christ says: "I am the good shepherd. The good shepherd gives His life for the sheep." Notice in John 17:9 that Jesus refuses to pray for those who God has not given him!
John 3:16 makes it plain that whilst God sent his son because he loved the world, all those who believe will experience salvation.
So, for whom did Jesus die - did he die for everyone or did his die with a specific group in mind.
Does the death of Jesus achieve what it set out to do?
What does the bible say the death of Jesus achieved?
Roman 5:19 "For as by one man's disobedience many were made sinners, so also by one Man's obedience many will be made righteous."
Notice that Paul doesn't say the death of Jesus makes some potentially righteous, he says it actually makes them righteous.
The free offer of the gospel
The question assumes that God is extending his gift of salvation universally, however that is presupposition that needs biblical support. From the very beginning God seems to be selective, he selects Abraham, then he selects Isaac (rejecting Ishmael) then Jacob, then the Jews - did he offer the same opportunities to Nahor, Ishmael, Esau, and the Canaanites?
What about people alive today in remote part of the world, places where the gospel has not yet reached? Does God extend his gift of salvation to them, if so, how does it do it?