Matthew 5

For I tell you, unless your righteousness exceeds that of the scirbes and Pharisees, you will never enter the kingdom of heaven.

Matthew 5:20

This was a very hard teaching for the Jewish in that time, when everyone thought the Pharisees were the 'most righteous' group of people in the world, whereas Jesus told people that your righteousness needs to exceeds the 'best of the best'. What Jesus wanted to convey to us was that sin is not about doing 'wrongs', but how we truly feel and think deep in our heart, as he continued to say:

21 "You have heard that it was said to those of old, 'You shall not murder; and whoever murders will be liable to judgment.' 22But I say to you that everyone who is angry with his brother will be liable to judgment; whoever insult his brother will be liable to the council; and whoever says, 'You fool!' will be liable to the hell of fire. 23 So if you are offering your gift at the altar and there remember that your brother has something against you, 24leave your gift there before the altar and go. First be reconciled to your brother, and then come and offer your gift.

Matthew 5:21-24

Jesus teaches us that what's important is not about how much you are going to offer at the altar, but what you are going to do to reconcile the relationship with your brother/sister who has done somethings against you.
In v24, "First be reconciled to your brother, and then come and offer your gift." Here I guess it also indicates that we should first forgive our brother/sister before coming back to the Lord. And indeed if we want to be forgiven by the Lord and receive His grace, we should learn from Him and do the same.

For it is better that you lose one of your members than that your whole body be thrown into hell. 30 And if your right hand causes you to sin, cut it off and throw it away. For it is better that you lose one of your members than that your whole body go into hell.

5:26-30

As brothers and sisters in Christ, we are usually described as different parts of the same body and everyone has different purposes but the same 'head'. Keeping in mind this metaphor, how applicable is this to the situation where there is a person in Christ who has sinned. Is it better to 'cut it off and throw it away', or to help him/her to come back to Christ?

5:16In the same way, let your light shine before others, so that they may see your good works and give glory to your Father who is in heaven.

AND

6:3But when you give to the needy, do not let your left hand know what your right hand is doing, 4so that your giving may be in secret.

The above verses seem to be contradicting. After collaborating with Ivan, we came up with the following conclusion: when we are helping others, we should do it with our heart and truly love others, but not because of a particular 'purpose'. To illustrate this, we came up with an example:
As people give food to the needy, we do not go everywhere and tell people we will do this 'good thing' because we are Christians. We do not purposefully give food to people because we want to 'make' them Christians, but because of love we do all those. And due to love (from Christ), people will see us as 'light' and hope of the world, and therefore we can share the gospel to others. Therefore the reason for helping other should not be mainly due to sharing the gospel, but the key reason is because of love...... (does it make sense....?)