If you look at your arm, what do you see? Skin, hair, moles, freckles, etc. Now put a magnifying glass to it or take a skin sample and look at it under a microscope. There is more to it than you see with the naked eye. I believe that's what Jesus did with the law in the Sermon on the Mount. He magnified it.
If your mindset is that life in Christ is all about doing this and doing that and following rules and principles, then you will naturally look at the Sermon on the Mount as a set of teachings that show Christians how to live. But when you hear the Sermon on the Mount, what you're really seeing (hearing) is a magnified view of the Law. Years before Paul referred to the law as "the ministry of death" and "the ministry of condemnation," Jesus was putting it under a microscope to show how it really was "against us" and "contrary to us," as Paul would also later say.
"It's not only 'you shall not commit adultery,' but if you look at someone with lust you've already committed adultery in your heart" (paraphrase).
"You shall be perfect as your Father in heaven is perfect."
"Unless your righteousness exceeds that of the scribes and Pharisees, you shall by no means enter the kingdom of heaven."
"If your right eye causes you to sin, or your right hand causes you to sin, pluck it out, cut if off..."
"If you do not forgive others their trespasses, neither will your Father forgive your trespasses."
Is this really the Christian life that Jesus is talking about?
He was speaking to those who were under the law. One reason the law was given was "that every mouth may be stopped" (Rom 3:19). If a person thought they had a handle on the law and righteous living, this was definitely a mouth-stopping sermon!
Rather than a teaching on principles for Christian living, is not the Sermon on the Mount actually a series of Jewish teachings that precede the Cross and all that was accomplished there, and a series of teachings that showed what people must die to in order to truly enter into Life?
Rom 7:4 Therefore, my brethren, you also have become dead to the law through the body of Christ, that you may be married to another — to Him who was raised from the dead, that we should bear fruit to God.
In order to "bear fruit to God," one must die to self-effort. One must die to trying to "be perfect as your Father in heaven is perfect." One must die from trying to have their righteousness exceed that of the scribes and Pharisees. One must realize that they are an adulterer, whether or not they've ever committed the physical act, and one must then see that the only way to end that is to die and to be born again - to become a new creation.