How do Calvinists argue that limited atonement is just and merciful?

Jesus declares some simple realities. Unless you are born from on high you will not see God. All are doomed to judgement and destruction.

But seek first his kingdom and his righteousness, and all these things will be given to you as well. Matthew 6:33

All things have been committed to me by my Father. No one knows the Son except the Father, and no one knows the Father except the Son and those to whom the Son chooses to reveal him. Matthew 11:27

A few who are predestined by God to be saved will be saved. But it is all framed in free will and free choice. Jesus emphasised everything is under Gods sovereign will, and those who respond respond to His grace and love. The how and why is a mystery, but plucked like coals from a fire are His chosen.

When the Gentiles heard this, they were glad and honored the word of the Lord; and all who were appointed for eternal life believed. Acts 13:48

So the message is preached, the seed sown and some turn, repent and believe.

What has value that which responds and gains life or that which rejects and keeps the life they have instead of walking into new life that Jesus holds out.

The spiritual dilemma is without revelation how can a heart respond to God, because a dead heart is dead. But praise the Lord some do respond to His calling. And the atonement is for those who are saved, because atonement only applies to those who by faith reach out to God. We see only a small part. So many in Israel saw Jesus, heard His words, yet rejected Him, just like the rich young ruler.

It is grace that any are saved, yet some believe man is capable of seeing what Jesus offers, without Him reaching out to give life. I have witnessed to people who even have gone to church their whole lives yet their hearts are cold and lock out the love Jesus offers.

In my theology, they go to be destroyed not tortured, to have their chaff burnt up, because all they possessed in their lives was the shell and not the essence. It is judgement on what they held more valuable than seeking His face, but it is the justice they themselves have passed on their own lives.

John Piper "“Christ died for everyone, but not everyone in the same way.”

Isn’t Unlimited Atonement More Glorious Than Limited Atonement?

The general idea is that limited atonement is just in that it shows justice to some (the damned) and merciful in that it shows mercy to some (the saved). Conversely, it does not show salvific mercy to some (the damned) and it does not show damning justice to some (the saved).

So under this understanding of justice and mercy, God is seen to be both fully just and fully merciful. These two attributes are uniquely expressed in God's decision to give the damned what they deserve and to not give the saved what they deserve.

I am a Calvinist and I will try to answer this difficult but explainable question in my thoughts and belief in scripture.
Until we can see man's total depravity and God's total sovereignty we will never be able to understand God's just means so please bear with me.

Few really believe in the complete ruin and total depravity of man. Those who speak of man's "free will" and insist upon his inherent power to either accept or reject the Savior, only voices their misunderstandings of the real condition of Adam's fallen children. So, if few believe that the condition of the sinner is entirely hopeless, there are fewer who really believe in the absolute sovereignty of God. ~ A.W. Pink

What do we mean by the expression "the sovereignty of God"? We mean the greatness of God, the subject of God as king, and the God-hood of God. Basically, God is God. To say God is sovereign is to assert that He is the Most High, doing "according to His will in His army of heaven, and among the inhabitants of the earth: and none can stay His hand, or say unto Him, What doest thou"? (Daniel 4:35).
To say that God is sovereign is to declare that He is the almighty and the owner of all power in heaven and earth. No one can defeat His plans, prevent His purpose or resist His will (Psalms 115:3). We proclaim that He is "The Governor among the nations" (Psalms 22:28), setting up kingdoms, overthrowing empires, and determining the pathway of the nation's as He decides is best and according to His wil,His desire and His good pleasure. To say that God is sovereign is to announce that He is the "Only Potentate, the King of kings and Lord of lords" (1Timothy 6:15). This is the real picture of God of the Bible.

The sovereignty of the God of the Bible is absolute, irresistible and infinite. We recognize His right to rule the universe just as He pleases. We affirm that His right is right of the potter over the clay to mold that clay into whatever form he chooses. He makes out of the same lump one vessel into honor and a other unto dishonor (Romans 9:21). As a Calvinist in my belief, I affirm that God is under no rule of laws outside of His own will and nature, that He is only accountable to Himself and under no obligation to give explanations of His business to anyone.
Sovereignty characterizes the whole being of God and His sovereign qualities.

For many centuries Israel was the only nation to whom God granted any special revelation of Himself:

• "Who in times past suffered all nations to walk in their own ways" (Acts 14:16
• "You only (Israel) have I known of all the families of the earth" (Amos 3:2)

• Consequently, as all other nations were without the preaching of God's word, they were strangers to the faith that cometh thereby (Romans 10:17)

  These nations were not only ignorant of God himself, but had no way to please Him as a man or to be accepted by Him. Did God provide salvation to those people? Has God changed? Does God not have the same rights as a sovereign today as He had then. Why was Israel God's chosen nation. Was it unjust for God to leave the other nations to themselves. NO! God is both just And merciful to whom He pleases and He wills. Why should we question God?

We as humans in our finite minds seem to think it is "all about us". Most all of us believe that God made all things but that He made all things for himself is not so widely believed. However, that God made us, not for our own sake, but for himself, not for our happiness, but for His own glory is repeatedly affirmed in Scripture.

"Thou art worthy, O Lord, to receive glory and honor and power: for thou hast created all things, and for thy pleasure they were created" (Revelation 4:11).

But Proverbs 16:4 even goes further. It clearly declares that the lord made the wicked for the day of evil; that was His purpose in giving them existence. God is our creator. Is is not fair that He and He alone uses us as he finds His very will, His good pleasure and glory. I say absolutely. He is God, I am not. All for His glory.

These are a few things that have come to mind as a Calvinist and as I think of your question. God is just and merciful to whom He chooses. This is of His makings, who am I to question? He is the "Almighty, the maker of all" all things to His glory. Before we can question why, we must understand who He is and His sovereignty and not forget who we are. I hope this can and will give you a better understanding of how most Calvinist see God.

KJV Bible, R.C. Sproul, A.W. Pink and John Mcarthur have all been contributors to my understanding of limited atonement. Understand who God is and His Holiness then the rest falls in place.