Does God forget our sin?  What does David mean when he says,

“I, even I, am he who blots out
    your transgressions, for my own sake,
    and remembers your sins no more.”                Isaiah 43:25  NIV

Or Isaiah:  “Lo, for my own welfare I had great bitterness;
It is You who has kept my soul from the pit of nothingness,
For You have cast all my sins behind Your back.”                                 Isaiah 38:17  NASB

“He will again have compassion on us;
He will tread our iniquities under foot.
Yes, You will cast all their sins
Into the depths of the sea.”                                                                           Micah 7:19    NASB

It is an incomprehensible blessing that we should be forgiven forever, we who deserve full, just punishment for our self-life.  His forgetting our sin is wonderful relief of our anxiety and sense of doom.  But our remembering our sin is also a powerful motivator in us.  The memory that we who are guilty have been released from those charges, our debt paid in full by Jesus, motivates us to godly living out of gratitude for this undeserved grace.
“Therefore, since we receive a kingdom which cannot be shaken, let us show gratitude, by which we may offer to God an acceptable service with reverence and awe; for our God is a consuming fire.”   Hebrews 12:28-29   NASB

“Therefore, I urge you, brothers and sisters, in view of God’s mercy, to offer your bodies as a living sacrifice, holy and pleasing to God—this is your true and proper worship.”       Romans 12:1  NIV

“For the love of Christ controls us, having concluded this, that one died for all, therefore all died; and He died for all, so that they who live might no longer live for themselves, but for Him who died and rose again on their behalf.”                   2 Corinthians 5:14-15