2 Corinthians 13:5
5 Examine yourselves, whether ye be in the faith; prove your own selves. Know ye not your own selves, how that Jesus Christ is in you, except ye be reprobates?
The above verse is one of the most common prooftexts for the doctrine that each Christian needs to examine their lives, and depending upon whether or not they see a nebulous amount of goodness, determine whether or not they are saved. This is then immediately applied to others as well, and becomes the basis for declaring people who live a relatively 'bad' life to be unsaved unbelievers, regardless of their profession.
Is this kind of "fruit inspecting", as it is often called, to determine whether one is saved, a legitimate application of this passage?
When examining the context of the verse in question, we can see that it is in response to the Corinthians' testing of the Apostle Paul, apparently brought about by some who were slandering him to them (2 Corinthians 10:2, 11:13):
2 Corinthians 13:3-7
3 Since ye seek a proof of Christ speaking in me, which to you-ward is not weak, but is mighty in you.
4 For though he was crucified through weakness, yet he liveth by the power of God. For we also are weak in him, but we shall live with him by the power of God toward you.
5 Examine yourselves, whether ye be in the faith; prove your own selves. Know ye not your own selves, how that Jesus Christ is in you, except ye be reprobates?
6 But I trust that ye shall know that we are not reprobates.
7 Now I pray to God that ye do no evil; not that we should appear approved, but that ye should do that which is honest, though we be as reprobates.
Notice above that the Apostle Paul, in response to the Corinthians "seeking a proof of Christ speaking in him", turns it around on them, telling them to instead "prove" themselves. Why does he do that?
Simply, it was because Paul was the one who originally preached the Gospel to the Corinthians, and they were saved under his preaching in the first place (1 Corinthians 15:1-2: 2 Corinthians 1:19):
1 Corinthians 4:15
15 For though ye have ten thousand instructers in Christ, yet have ye not many fathers: for in Christ Jesus I have begotten you through the gospel.
He "begot them through the Gospel", and yet, they are now questioning whether he is even a legitimate teacher. He's telling them, in our verse in question, that they themselves are his credentials of being a true teacher, as the faith that they believe was preached to them by him.
In other words, he's telling them to examine themselves, not because he doubts their Salvation - he doesn't, calling them "brethren" (2 Corinthians 1:8, 8:1, 13:11), and confirming their faith throughout the epistle (2 Corinthians 1:1-3, 1:21-22, 3:2-3, 6:14-17, 7:11, 8:7, 13:11-14, etc.) - but because he affirms it - "know ye not your own selves, how that Jesus Christ is in you" - and shares their same faith, meaning if he is disapproved (reprobate), they are disapproved, but if they are approved, then he is also:
2 Corinthians 10:7
7 Do ye look on things after the outward appearance? if any man trust to himself that he is Christ's, let him of himself think this again, that, as he is Christ's, even so are we Christ's.
Above is the Apostle Paul saying much the same thing as he does in the verse in question - if the Corinthians are Christ's, so is he.
It's also important to note that in the verse in question, he appears to be using a bit of the cleverness, bordering on sarcasm, that characterizes the end of this epistle (2 Corinthians 11:1, 11:8, 11:16-19, 12:13, 12:16, etc.).
In summary, this has nothing to do with the Apostle Paul issuing some vague command for the Corinthian church to look at their conduct to see whether or not they are "truly" saved. It has to do with the fact that they were his children in the faith, and how silly it was to treat him as suspiciously as they were, in light of that fact. He feels compelled to defend his apostleship throughout much of the epistle, because of this occasional suspicious attitude toward him (2 Corinthians 12:11-12, 12:15, etc.).
The common twisting of this verse is a case in point of why Scripture needs to be read in context, as it would be hard to come away with the works Salvation interpretation of this passage if one reads the whole of the letter that it's in.