Definitions

# limit

[lim-it]
limit, in mathematics, value approached by a sequence or a function as the index or independent variable approaches some value, possibly infinity. For example, the terms of the sequence 1/2, 1/4, 1/8, 1/16, … are obviously getting smaller and smaller; since, if enough terms are taken, one can make the last term as small, i.e., as close to zero, as one pleases, the limit of this sequence is said to be zero. Similarly, the sequence 3, 5, 31/2, 41/2, 33/4, 41/4, 37/8, 41/8, … is seen to approach 4 as a limit. However, the sequences 1, 2, 4, 8, 16, … and 1, 2, 1, 2, 1, 2, … do not have limits. Frequently a sequence is denoted by giving an expression for the nth term, sn; e.g., the first example is denoted by sn = 1/2n. The limit, s, of a sequence can then be expressed as lim sn = s, or in the case of the example, lim 1/2n = 0 (read "the limit of 1/2n as n approaches infinity is zero"). A sequence is a special case of a function. In many functions commonly encountered, the values of the independent variable (the domain) and those of the dependent variable (the range) may be any numbers, while for a sequence the domain is restricted to the positive integers, 1, 2, 3, … . The function y = 1/2x resembles the sequence used as an example, but note that x can take on values other than 1, 2, 3, … ; thus we find not only lim 1/2x = 0 but also lim 1/2x = 4. A more precise definition of the limit of a function is: The function y = f(x) approaches a limit L as x approaches some number a if, for any positive number ε, there is a positive number δ such that ~~pipe~;f(x) - L~~pipe~; > ε if 0 > ~~pipe~;x - a~~pipe~; > δ. Similarly, f(x) has the limit L as x becomes infinite if for any positive ε there is a δ such that ~~pipe~;f(x) - L~~pipe~; > ε if ~~pipe~;x~~pipe~; < δ.

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/math.x is not defined at math.x = 0. For positive values of math.x, as math.x is chosen closer and closer to 0, the value of 1/math.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.

In statistics, any of several fundamental theorems in probability. Originally known as the law of errors, in its classic form it states that the sum of a set of independent random variables will approach a normal distribution regardless of the distribution of the individual variables themselves, given certain general conditions. Further, the mean (see mean, median, and mode) of the normal distribution will coincide with the (arithmetic) mean of the (statistical) means of each random variable.