When f(x) is continuous at x = a, limit and function value at a are the same.
In your example, where f(x) has a "perforation", the limit represents the value which would make f(x) continuous there, it would "fill the hole".
Have you gotten a rigorous definition of a limit (epsilon-delta perhaps?) and if so, do you fully understand it?
As for your last question, if f(x) is defined on a neighborhood of x = a, then the limit of f(x) for x approaching a exists if and only if upper and lower limit (or "right" and "left") exist and are equal. Use this on f(x) = 1/x to show that the limit doesn't exist.