I'm not sure what you're looking for...?
The limit, as x approaches some value "a", of f(x) is nothing more than what the value of f(a) ought reasonably to be, assuming the limit exists.
In some cases, the limit does not exist, such as for f(x) = 1/x when x approaches zero.
In other cases, the limit exists, but not the functional value, such as for f(x) = [(x + 1)(x - 2)]/(x - 2) when x approaches 2. Other than for x = 2, this function is the same as g(x) = x + 1, so the limit at x = 2 is g = 3. But f(x) is not actually defined for x = 2. The limit exists, but the function doesn't actually take on that value.
In still other cases, the limit exist, and the functional value exists, such as for f(x) = x.
And then you have the cases where the limit from one side exists, but the other limit does not, or is not the same value. A piecewise function is a good example of this:
Clearly, each "half" has a limit as x approaches zero: from the left, the function "ought" to take on the value -1 (and it does); from the right, the function "ought" to take on the value 1 (but it doesn't, because f(0) = -1). Each one-sided limit exists, but "the" limit does not, because the two one-sided limits don't agree.
Does that help at all...?