LIMIT

mathematical concept based on the idea of closeness, used primarily to assign values to certain functions at points where no values are defined, in such a way as to be consistent with nearby values. For example, the function (x2 - 1)/(x - 1) is not defined when x is 1, because division by zero is not a valid mathematical operation. For any other value of x, the numerator can be factored and divided by the (x - 1), giving x + 1. This is equivalent to the quotient for all values of x except 1, in which it is equal to 2, in contrast to the quotient that has no value. This value of 2 is then assigned to the function (x2 - 1)/(x - 1) not as its value when x equals 1, but as its limit when x approaches 1. One way of defining the limit of a function f (x) at a point x0, written as is by the following: if there is a continuous (unbroken) function g(x) such that g(x) = f (x) in some interval around x0, except possibly at x0 itself, then The following more basic definition of limit, independent of the concept of continuity, can also be given: if, for any desired degree of closeness e, one can find an interval around x0 so that all values of f (x) calculated here differ from L by an amount less than e (i.e., if |x - x0|