Caspar Olevianus, A Firm Foundation, Questions 141-142 (translated by Lyle D. Bierma [more info])
Q. Since there is nothing more difficult to believe than the forgiveness of sins, give me some reasons or grounds on which to base (or establish) our belief that believers are certainly forgiven of their sins.
A. The reason and ground for our certainty of forgiveness of sins through Christ is the promise and oath of God, confirmed in actual fact in the death of Christ, as explained in the preceding articles about Christ. There is no condition that we have to keep the commandments; it is a free gift appropriated through faith or trust in the merit of Christ, without any merit of works. Faith must look directly at this voluntary promise and oath of God in Jesus Christ made for the sake of His merit (for in Him all the promises of God have their "Yes" and "Amen" [2Co 1.20], as Heb 6.17-20 says.
Q. Are we to believe, however, that our sins are forgiven us in such a way that there is no more sin in us?
A. No. But even though there is sin in us now and until our death, I believe that it is not imputed to us but fully pardoned. That is why St. Paul and the prophets declared as saved not those who have no sin but those whose sin, while real, is covered (Rom 4.7; Ps 32.1).
[Questions & Answers Copyright 1995 Lyle D. Bierma]
It is indisputable that Jesus claimed the ability to forgive sin. Both Matthew and Luke record this (Mt 9.1-8 || Lu 5.17-25). Here's the account from Matthew. I've made a few parts bold to emphasize them.
And getting into a boat he crossed over and came to his own city. And behold, some people brought to him a paralytic, lying on a bed. And when Jesus saw their faith, he said to the paralytic, “Take heart, my son; your sins are forgiven.” And behold, some of the scribes said to themselves, “This man is blaspheming.” But Jesus, knowing their thoughts, said, “Why do you think evil in your hearts? For which is easier, to say, ‘Your sins are forgiven,’ or to say, ‘Rise and walk’? But that you may know that the Son of Man has authority on earth to forgive sins”—he then said to the paralytic—“Rise, pick up your bed and go home.” And he rose and went home. When the crowds saw it, they were afraid, and they glorified God, who had given such authority to men. (Mt 9.1-8, ESV)
Yet this made the religious authorities of the day (scribes & Pharisees) nervous, because they immediately understood what it was that Jesus was claiming in this action. Jesus wasn't forgiving sin as one man forgives another man who has wronged him. The paralytic had not wronged Jesus. Jesus here is forgiving sin, the underlying "wrong-being" within man. Only God can address this wrong-being. Jesus knew this. So did the scribes, which is why they reacted the way they did. In doing this, Jesus was asserting his claim to be God. Yet Jesus went further and made the claim explicit: " ... that you may know that the Son of Man has authority on earth to forgive sins".
Logically, the statements work like this:
Only God can forgive sin
Jesus forgives sin
Therefore Jesus must be God
At least, that's the math that the scribes did. And the logic is correct. Without forgiveness of sin, salvation is not possible. If the condition of sin within men is not addressed, man cannot approach God. Forgiveness of sin is essential. If Jesus really does forgive sin, then he really must be God. The apostles understood this in their early preaching. Peter understood it at Pentecost:
Now when they heard this they were cut to the heart, and said to Peter and the rest of the apostles, “Brothers, what shall we do?” And Peter said to them, “Repent and be baptized every one of you in the name of Jesus Christ for the forgiveness of your sins, and you will receive the gift of the Holy Spirit. For the promise is for you and for your children and for all who are far off, everyone whom the Lord our God calls to himself.” And with many other words he bore witness and continued to exhort them, saying, “Save yourselves from this crooked generation.” So those who received his word were baptized, and there were added that day about three thousand souls. (Ac 2.37-41, ESV)
The forgiveness of sin is central to the apostolic preaching because it is what allows us to approach God through our mediator Jesus Christ. No longer do priests need to propitiate God with yearly animal sacrifices to temporarily cover the sin of the people; that practice is dead. The curtain is ripped and removed, the Holy of Holies is accessible to all (cf. Heb 10.1-18). The sacrifice of Jesus Christ provides complete, permanent forgiveness of sin. Through Him, because of this forgiveness, we are able to approach God. Again, here's an account of Peter from Acts:
And when they had brought them, they set them before the council. And the high priest questioned them, saying, “We strictly charged you not to teach in this name, yet here you have filled Jerusalem with your teaching, and you intend to bring this man’s blood upon us.” But Peter and the apostles answered, “We must obey God rather than men. The God of our fathers raised Jesus, whom you killed by hanging him on a tree. God exalted him at his right hand as Leader and Savior, to give repentance to Israel and forgiveness of sins. And we are witnesses to these things, and so is the Holy Spirit, whom God has given to those who obey him.” (Ac 5.27-32, ESV)
Peter was compelled to preach this forgiveness despite the circumstances because he knew it to be true. He had experienced it. He witnessed it first-hand. He had (most likely) seen the paralytic forgiven and healed.
Though we are forgiven if we claim Jesus Christ as our Lord and Savior, this does not mean that we are immediately made perfect and will cease from sinning as if by magic. Will our lives be changed? Yes, they will. We should seek to serve Christ our Savior and come ever more familiar with Him and his love for us. But this does not mean we will stop sinning; it means that our sin is pardoned and "covered". Consider Psalm 32:
1 Blessed is the one whose transgression is forgiven, whose sin is covered.
2 Blessed is the man against whom the Lord counts no iniquity, and in whose spirit there is no deceit.
3 For when I kept silent, my bones wasted away through my groaning all day long.
4 For day and night your hand was heavy upon me; my strength was dried up as by the heat of summer. Selah
5 I acknowledged my sin to you, and I did not cover my iniquity; I said, “I will confess my transgressions to the Lord,” and you forgave the iniquity of my sin. Selah
6 Therefore let everyone who is godly offer prayer to you at a time when you may be found; surely in the rush of great waters, they shall not reach him.
7 You are a hiding place for me; you preserve me from trouble; you surround me with shouts of deliverance. Selah
(Ps 32.1-7, ESV)
Paul knew this as well. Paul knew that he had been forgiven and that he served a risen Lord. Yet in his day-to-day life, he found that he still struggled with sin. Though "covered", he still had to wage war against it in his body, in his daily life. He wasn't perfect, but he did his best to fight the sin he encountered in his walk. Here's what he writes in Romans:
So I find it to be a law that when I want to do right, evil lies close at hand. For I delight in the law of God, in my inner being, but I see in my members another law waging war against the law of my mind and making me captive to the law of sin that dwells in my members. Wretched man that I am! Who will deliver me from this body of death? Thanks be to God through Jesus Christ our Lord! So then, I myself serve the law of God with my mind, but with my flesh I serve the law of sin. (Ro 7.21-25, ESV)
Olevianus sums it up this way in Question 143:
Is this, then, a summary of the article on the forgiveness of sins: You believe that the church, the body of Christ, and all her members possess in this life a forgiveness that is not uncertain or temporary but certain, lasting and eternal; that it is a forgiveness not just sof some sins but of all the sins whith which they have to struggle daily; that there is in the church, therefore, no more condemnation, just as if there were no more sin and death; and that believers have peace with God and thus true salvation?