 limit

—limitable, adj. —limitableness, n./lim"it/, n.1. the final, utmost, or furthest boundary or point as to extent, amount, continuance, procedure, etc.: the limit of his experience; the limit of vision.2. a boundary or bound, as of a country, area, or district.3. Math.a. a number such that the value of a given function remains arbitrarily close to this number when the independent variable is sufficiently close to a specified point or is sufficiently large. The limit of 1/x is zero as x approaches infinity; the limit of (x  1)^{2} is zero as x approaches 1.b. a number such that the absolute value of the difference between terms of a given sequence and the number approaches zero as the index of the terms increases to infinity.c. one of two numbers affixed to the integration symbol for a definite integral, indicating the interval or region over which the integration is taking place and substituted in a primitive, if one exists, to evaluate the integral.4. limits, the premises or region enclosed within boundaries: We found them on school limits after hours.5. Games. the maximum sum by which a bet may be raised at any one time.6. the limit, Informal. something or someone that exasperates, delights, etc., to an extreme degree: You have made errors before, but this is the limit.v.t.7. to restrict by or as if by establishing limits (usually fol. by to): Please limit answers to 25 words.8. to confine or keep within limits: to limit expenditures.9. Law. to fix or assign definitely or specifically.[132575; ME lymyt < L limit (s. of limes) boundary, path between fields]Syn. 2. confine, frontier, border. 8. restrain, bound.
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IMathematical concept based on the idea of closeness, used mainly in studying the behaviour of functions close to values at which they are undefined.For example, the function 1/x is not defined at x = 0. For positive values of x, as x is chosen closer and closer to 0, the value of 1/x begins to grow rapidly, approaching infinity as a limit. This interplay of action and reaction as the independent variable moves closer to a given value is the essence of the idea of a limit. Limits provide the means of defining the derivative and integral of a function.II(as used in expressions)limited obligation bond* * *
mathematical concept based on the idea of closeness, used primarily to assign values to certain functions (function) at points where no values are defined, in such a way as to be consistent with nearby values. For example, the function (x^{2} − 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. Thus, this quotient is equal to 2 for all values of x except 1, which has no value. However, 2 can be assigned to the function (x^{2} − 1)/(x − 1) not as its value when x equals 1 but as its limit when x approaches 1. See analysis: Continuity of functions (analysis).One way of defining the limit of a function f(x) at a point x_{0}, written asis by the following: if there is a continuous (continuity) (unbroken) function g(x) such that g(x) = f(x) in some interval around x_{0}, except possibly at x_{0} itself, thenThe following morebasic definition of limit, independent of the concept of continuity, can also be given:if, for any desired degree of closeness ε, one can find an interval around x_{0} so that all values of f(x) calculated here differ from L by an amount less than ε (i.e., if x − x_{0} < δ, then f (x) − L < ε). This last definition can be used to determine whether or not a given number is in fact a limit. The calculation of limits, especially of quotients, usually involves manipulations of the function so that it can be written in a form in which the limit is more obvious, as in the above example of (x^{2} − 1)/(x − 1).Limits are the method by which the derivative, or rate of change, of a function is calculated, and they are used throughout analysis as a way of making approximations into exact quantities, as when the area inside a curved region is defined to be the limit of approximations by rectangles.* * *
Universalium. 2010.