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...?