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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

Mathematical concept based on the idea of closeness, used mainly in studying the behaviour of functions close to values at which they are undefined. For example, the function 1/*math.x* is not defined at *math.x* = 0. For positive values of *math.x*, as *math.x* is chosen closer and closer to 0, the value of 1/*math.x* begins to grow rapidly, approaching infinity as a limit. This interplay of action and reaction as the independent variable moves closer to a given value is the essence of the idea of a limit. Limits provide the means of defining the derivative and integral of a function.

Learn more about limit with a free trial on Britannica.com.

Encyclopedia Britannica, 2008. Encyclopedia Britannica Online.

In statistics, any of several fundamental theorems in probability. Originally known as the law of errors, in its classic form it states that the sum of a set of independent random variables will approach a normal distribution regardless of the distribution of the individual variables themselves, given certain general conditions. Further, the mean (*see* mean, median, and mode) of the normal distribution will coincide with the (arithmetic) mean of the (statistical) means of each random variable.

Learn more about central limit theorem with a free trial on Britannica.com.

Encyclopedia Britannica, 2008. Encyclopedia Britannica Online.

Minimum distance at which a large natural satellite can orbit its primary body without being torn apart by tidal forces. If satellite and primary are of similar composition, the theoretical limit is about 2.5 times the radius of the larger body. The rings of Saturn, for example, lie inside Saturn's Roche limit and may be the debris of a demolished moon. The limit was first calculated by the French astronomer Édouard Roche (1820–53) in 1850.

Learn more about Roche limit with a free trial on Britannica.com.

Encyclopedia Britannica, 2008. Encyclopedia Britannica Online.

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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