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In Mathematics , the concept of a "limit" is used to describe the Behavior of a Function as its Argument either "gets close" to some point, or as it becomes arbitrarily large; or the behavior of a Sequence 's elements as their Index increases indefinitely. Limits are used in Calculus and other branches of Mathematical Analysis to define Derivative s and Continuity .

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 .


LIMIT OF A FUNCTION

See Also: Limit of a function



Suppose ƒ(''x'') is a real function and ''c'' is a real number. The expression:

: \lim_{x o c}f(x) = L

means that ƒ(''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 ƒ of ''x'', as ''x'' approaches ''c'', is ''L''". Note that this statement can be true even if \scriptstyle f(c)
eq L. Indeed, the function ƒ(''x'') need not even be defined at ''c''. Two examples help illustrate this.

Consider ''f''(''x'') = ''x''/(''x''2 + 1) as ''x'' approaches 2. In this case, ''f''(''x'') is defined at 2 and equals its limit of 0.4:

As ''x'' approaches 2, ƒ(''x'') approaches 0.4 and hence we have \scriptstyle \lim_{x o 2}f(x)=0.4.
In the case where \scriptstyle f(c) = \lim_{x o c} f(x), ƒ is said to be Continuous at ''x'' = ''c''. But it is not always the case. Consider

:g(x)=\left\{\begin{matrix} rac{x}{x^2+1}, & \mbox{if }x
e 2 \ \ 0, & \mbox{if }x=2. \end{matrix} ight.

The limit of ''g''(''x'') as ''x'' approaches 2 is 0.4 (just as in ƒ(''x'')), but \scriptstyle \lim_{x o 2}g(x)
eq g(2); ''g'' is not continuous at ''x'' = 2.

Or, consider the case where ƒ(''x'') is undefined at ''x'' = ''c''.

: f(x) = rac{x - 1}{\sqrt{x} - 1}

In this case, as ''x'' approaches 1, ''f''(''x'') is undefined at ''x'' = ''1'' but the limit equals 2:

Thus, ''f''(''x'') can be made arbitrarily close to the limit of 2 just by making ''x'' sufficiently close enough to 1.


Formal definition

Karl Weierstrass formally defined a limit as follows:

Let ''f'' be a function defined on an Open Interval containing ''c'' (except possibly at ''c'') and let ''L'' be a Real Number .

: \lim_{x o c}f(x) = L

means that