We have two questions. I will answer them in the order you have posted them.
1. If f(2) = 4, can we conclude anything about the limit of the function as x approaches 2.
The answer to this question is 'No'. The reason is pretty simple, the limit of a function doesn't depend on the actual value of the function at a point. This can be understood very well with an example of a function with a jump discontinuity. At a jump discontinuity, the value of function may exist, however limit doesn't exist.
2) If the limit of f(x) as x approaches 2 is 4, can we conclude anything about f(2)?
The answer to this question is 'No'. The reason is, the value of a function doesn't depend on limit of the function at a point. This can be understood very well with a question where we have a hole. A rational function with a hole is a very good example that will clarify this question. Limit of a function at a point where we have a hole in the function will exist, however, value of function will not exist.