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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 *n*th term, *s*_{n}; e.g., the first example is denoted by *s*_{n} = 1/2^{n}. The limit, *s,* of a sequence can then be expressed as lim *s*_{n} = *s,* or in the case of the example, lim 1/2^{n} = 0 (read "the limit of 1/2^{n} 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* = 1/2^{x} 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 1/2^{x} = 0 but also lim 1/2^{x} = 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.

The Columbia Electronic Encyclopedia Copyright © 2004.

Licensed from Columbia University Press

Licensed from Columbia University Press

A limit can be:## See also

- Limit (mathematics), including:
- Limit of a function
- Limit of a sequence
- One-sided limit
- Limit superior and limit inferior
- Limit of a net
- Limit point
- Limit (category theory)
- A constraint (mathematical, physical, economical, legal, etc.) in the form of an inequality, such as:
- An extreme value or boundary, such as:
- High frequency limit
- A limit order is a type of order to buy a security at no more (or sell at no less) than a specific price on an exchange.
- Other uses, such as:
- Limit (music) in just intonation
- In BDSM, limits are activities that a partner feels strongly about, and to which special attention must be paid.
- The Limit, a 1980s band

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Last updated on Friday August 01, 2008 at 08:35:33 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Friday August 01, 2008 at 08:35:33 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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