The expected value of the function as dictated by the points to the left of a point defines the left hand limit of the function at that point. Similarly the right hand limit.

Limit of a function at a point is the common value of the left and right hand limits, if they coincide.

For a function f and a real number a, \lim_{x \to a} {f(x)} and f(a) may not be same (In fact, one may be defined and not the other one).