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In mathematics, a limit is the value that a function or sequence "approaches" as the input or index approaches some value. Limits are essential to calculus (and mathematical analysis in general) and are used to define continuity, derivatives, and integrals. The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory. In formulas, a limit is usually written as : and is read as "the limit of ''f'' of ''n'' as ''n'' approaches ''c'' equals ''L''". Here "lim" indicates ''limit'', and the fact that function ''f''(''n'') approaches the limit ''L'' as ''n'' approaches ''c'' is represented by the right arrow (→), as in : == Limit of a function == (詳細はreal-valued function and is a real number. The expression : means that can be made to be as close to as desired by making sufficiently close to . In that case, the above equation can be read as "the limit of of , as approaches , is ". Augustin-Louis Cauchy in 1821, followed by Karl Weierstrass, formalized the definition of the limit of a function as the above definition, which became known as the (ε, δ)-definition of limit in the 19th century. The definition uses (the lowercase Greek letter ''epsilon'') to represent any small positive number, so that " becomes arbitrarily close to " means that eventually lies in the interval , which can also be written using the absolute value sign as .〔 The phrase "as approaches " then indicates that we refer to values of whose distance from is less than some positive number (the lower case Greek letter ''delta'')—that is, values of within either or , which can be expressed with . The first inequality means that the distance between and is greater than and that , while the second indicates that is within distance of .〔 Note that the above definition of a limit is true even if . Indeed, the function need not even be defined at . For example, if : then is not defined (see division by zero), yet as moves arbitrarily close to 1, correspondingly approaches 2: Thus, can be made arbitrarily close to the limit of 2 just by making sufficiently close to . In other words, This can also be calculated algebraically, as for all real numbers . Now since is continuous in at 1, we can now plug in 1 for , thus . In addition to limits at finite values, functions can also have limits at infinity. For example, consider : * ''f''(100) = 1.9900 * ''f''(1000) = 1.9990 * ''f''(10000) = 1.99990 As becomes extremely large, the value of approaches 2, and the value of can be made as close to 2 as one could wish just by picking sufficiently large. In this case, the limit of as approaches infinity is 2. In mathematical notation, : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Limit (mathematics)」の詳細全文を読む スポンサード リンク
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