"Limited" may first strike you as improperly weak or incomplete. However, both Arminians and Calvinists agree that not all men will be saved as the result of Christ's redeeming work or atonement.

For both the Arminian and the Calvinist, the atonement will be limited in one sense or in another sense. "Either the atonement was limited in that it was designed to secure salvation for certain sinners but not for others (Calvinist position), or it was limited in that it was not intended to secure salvation for any, but was designed only to make it possible for God to pardon sinners on the condition that they believe (Arminian position)." (I add parenthetically the name to each position. Citation at the end of next paragraph.)

"In other words, one must limit its design either in extent (it was not intended for all) or effectiveness (it did not secure salvation for any). As Boettner so aptly observes, for the Calvinist, the atonement "is like a narrow bridge which goes all the way across the stream; for the Arminian it is like a great wide bridge that goes only half-way across'." David N. Steele and Curtis Thomas, The Five Points of Calvinism - Defined Defended Documented, at page 39 and 40, Presbyterian & Reformed Publishing, 1963 quoting Boettner, Loraine. The Reformed Doctrine of Predestination, Presbyterian & Reformed Publishing, 1992.

1. Notice the golden chain of salvation in Romans 8: 29-30:
“For those whom He foreknew, He also predestined to become conformed to the image of His Son, so that He would be the firstborn among many brethren; and these whom He predestined, He also called; and these whom He called, He also justified; and these whom He justified, He also glorified.”

2. Likewise, notice how every spiritual blessing, including redemption, is given to those who are “predestined” also known as “chosen before the foundation of the world”, in Ephesians 1: 3-8a:
“Blessed be the God and Father of our Lord Jesus Christ, who has blessed us with every spiritual blessing in the heavenly places in Christ, just as He chose us in Him before the foundation of the world, that we would be holy and blameless before Him.
In love He predestined us to adoption as sons through Jesus Christ to Himself, according to the kind intention of His will, to the praise of the glory of His grace, which He freely bestowed on us in the Beloved.
In Him we have redemption through His blood, the forgiveness of our trespasses, according to the riches of His grace which He lavished on us.”

The Arminian great wide bridge only goes half-way across the stream. It is a great wide bridge in the sense that everyone is given an opportunity to be saved, but salvation is not secured for anyone in particular. According to the Arminian, atonement was not intended to secure salvation for any, but was designed only to make it possible for God to pardon sinners on the condition that they believe. Talk about a leap of faith! The Arminian places responsibility on man to get enough faith (take a big enough leap of faith) and keep enough faith until death.

In contrast, the Calvinist sees Scripture as showing that God even gives faith.
The Calvinist bridge may be a narrow bridge in the sense that it is only for the elect, but it is a complete bridge that goes all the way across the stream.   Faith is included in the bridge, unlike in the Arminian bridge. Faith is a work of God, and faith is a gift of God
1.   John 6: 28-28: “Therefore they said to Him, "What shall we do, so that we may work the works of God?" Jesus answered and said to them, "This is the work of God, that you believe in Him whom He has sent."
2.   Ephesians 2: 8-9: “For by grace you have been saved through faith; and that not of yourselves, it is the gift of God; not as a result of works, so that no one may boast.”
3.   2 Peter 1:1.: "to those who have received a faith of the same kind as ours ...".

But, what about the universal passages?
"One reason for the use of these expressions was to correct the false notion that salvation was for the Jews alone. Such phrases as "the world", "all men", "all nations", and "every creature" were used by New Testament writers to emphatically correct this mistake. These expressions are intended to show that Christ died for all men without distinction (i.e., He died for Jews and Gentiles alike) but they are not intended to indicate that Christ died for all men without exception (i.e., He did not die for the purpose of saving each and every lost sinner). There are other passages which speak of His saving work in definite terms and show that it was intended to infallibly save a particular people, namely those given to Him by the Father." David N. Steele and Curtis Thomas, The Five Points of Calvinism - Defined Defended Documented, at page 46.

The reader of Scripture can understand that the use of the word "all" does not necessarily mean everybody existing in the world at that time. For example, John 12:19b records the Pharisees as saying: "Behold, the world is gone after him." It is obvious to the reader that not everybody existing in the world at that time was following Jesus; the reader will understand that the Pharisees were observing many following Jesus. As another example, Luke 2:3 records: "And all were proceeding to register for the census, everyone to his own city." Again, it is obvious to the reader that not everybody existing in the world at that time was proceeding to register; the reader will understand that many were proceeding to register.

We also need to consider the context of any universal term such as "all", namely the people to whom the New Testament writer is writing. The writer may be writing to believers, to the elect. For example, Peter was writing "to those who have received a faith of the same kind as ours ...". 2 Peter 1:1. So, if we see a passage in 2 Peter that uses such a universal term as "all", we can understand that he is referring to all believers.

In another sense, the use of such universal words, such as “the world” and “all”, is used to represent people from every nation. God purchased with Christ’s blood men from every tribe and tongue and people and nation. “And they sang a new song, saying, ‘Worthy are You to take the book and to break its seals; for You were slain, and purchased for God with Your blood men from every tribe and tongue and people and nation.’ ” Revelation 5: 9.

Matthew 20:28: "... the Son of Man came not to be served but to serve, and to give life as a ransom for many." Note that it does not say that Christ gave his life as a ransom for all, but rather, “for many”.

Matthew 26:28: "for this is My blood of the covenant, which is poured out for many for forgiveness of sins." Note that it does not say that Christ poured out His blood for all, but rather, “for many”.

John 10:11-14: "I am the good shepherd. The good shepherd lays down his life for the sheep. ... "I am the good shepherd; and I know My own, and My own know Me,"
John 10:27: “My sheep hear My voice, and I know them, and they follow Me;”

Consider Ephesians 5:25-28 which I will quote and discuss at length:
"Husbands, love your wives, just as Christ also loved the church and gave Himself up for her, so that He might sanctify her, having cleansed her by the washing of water with the word, that He might present to Himself the church in all her glory, having no spot or wrinkle or any such thing; but that she would be holy and blameless.
So husbands ought also to love their own wives as their own bodies."
This is a peculiar love, a special love just for His church. Just as a husband should not love and give himself up for all women, but only his own wife, Christ loved the church and gave Himself up for the church, not for all the world.

In Christ's intercessory prayer in John 17, we see Christ praying for only those whom God gave Him, not for the world. "I ask on their behalf; I do not ask on behalf of the world, but of those whom Thou hast given Me; for they are Thine;" (verse 9).
Christ likewise manifested God’s name to the men whom God gave Him.
“I manifested Thy name to the men whom Thou gavest Me out of the world; Thine they were, and Thou gavest them to Me, and they kept Thy word.” (Verse 6).

Was Christ’s passion, suffering, and death a true substitute? For whom?
Christ was a true substitute in receiving punishment and only a substitute for us. We understand that Christ’s passion, suffering, and death was not for the world and did not take away the world’s sins.

If Christ died for the whole world, the whole world's sins would have been taken away. No one would go to hell. But rather, Christ’s passion, suffering, and death was only for us and only took away our sins. But “... he who does not obey the Son shall not see life, but the wrath of God abides on him.” John 3: 36.

When we believe that Christ died only for us, His elect, not for the world, this ‘Limited Atonement”, the third point of Calvinism, is so much more appreciated by us.
"But He was pierced through for our transgressions,
He was crushed for our iniquities;
The chastening for our well-being fell upon Him,
And by His scourging we are healed." Isaiah 53: 5. (Emphasis added.)

LIMITED ATONEMENT The Third of the Five Points of Calvinism: The “L” in T-U-L-I-P”

LIMITED ATONEMENT
The Third of the Five Points of Calvinism: The “L” in T-U-L-I-P”