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.

[1325-75; ME lymyt < L limit- (s. of limes) boundary, path between fields]

Syn. 2. confine, frontier, border. 8. restrain, bound.

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I

Mathematical 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)

central limit theorem

Roche limit

limited liability

Qantas Airways Limited

limited obligation bond

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▪ mathematics

      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 (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. 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 (x² −  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 x₀, written as

is by the following: if there is a continuous (continuity) (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 ε, one can find an interval around x₀ so that all values of f(x) calculated here differ from L by an amount less than ε (i.e., if |x −  x₀| < 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 (x² −  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.

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