"Nothing I have read in the Bible convinces me that God gives commandments only to show us that we cannot obey them. On the contrary, it seems clear enough that God intends for us to keep his commandments."
Please read Romans 1:18 thru Romans 3:20. Paul is showing us that it is impossible to keep the commandments until we are justified by Christ. Until we receive the Holy Spirit we cannot begin to keep them. Because prior to having Christ's righteousness imputed to us we only glorify ourselves and not God.
The epistle to the Romans amply supports my belief that God requires obedience to his commandments. Nowhere does Paul rescind the commandment of Jesus, "Be ye therefore perfect, even your Father which is in heaven is perfect" (Matthew 5:48). Nor does Paul teach that obedience to God's other commandments is unnecessary. Quite the contrary. He warns the saints, "that to whom ye yield yourselves servants to obey, his servants ye are." (Romans 6:16). Those who yield to sin, receive the wages of sin, which is death, not eternal life (Romans 6:23).