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 |f(x) − L| < ε if 0 < |x − a| < δ. Similarly, f(x) has the limit L as x becomes infinite if for any positive ε there is a δ such that |f(x) − L| < ε if |x| > δ.
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