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The concept of the "limit of a function" is further generalized to the concept of Topological Net , while the limit of a sequence is closely related to Limit and Direct Limit in Category Theory . Mathematics students usually first encounter limits in introductory Calculus classes, and understanding the detailed concept often presents a stumbling block. Readers seeking an introductory explanation might look at the Wikibooks Calculus section about limits {Link without Title} . This present article does have some elementary exposition but it's also about how limits are treated in more advanced branches of mathematics. LIMIT OF A FUNCTION Main article: Limit Of A Function Suppose ''f''(''x'') is a real function and ''c'' is a real number. The expression: : means that ''f''(''x'') can be made to be as close to ''L'' as desired by making ''x'' sufficiently close to ''c''. In that case, we say that "the limit of ''f''(''x''), as ''x'' approaches ''c'', is ''L''". Note that this statement can be true even if ''f''(''c'') ''L''. Indeed, the function ''f''(''x'') need not even be defined at ''c''. Two examples help illustrate this. Consider as ''x'' approaches 2. In this case, ''f''(''x'') is defined at 2 and equals its limit of 0.4: As ''x'' approaches 2, ''f''(''x'') approaches 0.4 and hence we have . In the case where , ''f'' is said to be Continuous at ''x'' = ''c''. But it is not always the case. Consider : The limit of ''g''(''x'') as ''x'' approaches 2 is 0.4 (just as in ''f''(''x'')), but ; ''g'' is not continuous at ''x'' = 2. Or, consider the case where ''f''(''x'') is undefined at ''x'' = ''c''. : In this case, as ''x'' approaches 1, ''f''(''x'') is undefined at ''x'' = ''1'' but the limit equals 2: Thus, ''x'' can get as close to ''1'', so long as it is not equal to ''1'', so that the limit of is ''2''. Formal definition A limit is formally defined as follows: Let be a function defined on an open interval containing (except possibly at ) and let be a real number. The statement : |
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