In a BDSM context, limits are those kinks that you just won't do. Often divided into two catagories: soft limits and hard limits.

Watersports, kids, non-consentual, breath play, and extreme pain are all the limits -I- have at this time. That could change in the future, however...

LM

In calculus and elsewhere, we encounter functions that seem to tend towards a certain value without necessarily ever reaching it. For example, we want to say that the expression ((x+dx)2-x2)/dx is tending towards 2x for small values of dx, even though the division is undefined when the denominator is zero. The problem then is how to reason about this without introducing "infinitly small" quantities, since they have a tendency to cause contradictions.

The modern definition is due to Karl Weierstrass. It is similar to a promise: no matter how small inexactness ε a devil's advocate is prepared to accept, we can always get so close that he is satisfied. The formulation of a satisfactory definition of limits was one of the key points in the 19th century project to recast calculus in the rigorous form known as analysis.

definition A function f(z) is said to have a limit limz→a f(z) = c ("the limit of f as z tends to a is c") if, for every number ε > 0, there exists a number δ > 0, such that |f(z)-c| < ε whenever 0 < |z-a| < δ.

The points to note are that the inequalities are strict, and that the value of f(a) is never used even if it is defined.

Examples, consider the limits of the following real functions as x tends to 0:

• The identity function f(x) = x tends to 0. Given any ε we can take δ = ε. Then if 0 < |x-0| < δ, we have |f(x)-0| = |x| < ε, as required.
• If we change the above function at a single point by letting f(x) = 42 when x = 0, f(x) = x otherwise, it will still tend to 0 by the same argument as above.
• The function f(x) = x sin(1/x) looks more complicated, but given any ε we can still take δ = ε. Since |sin(1/x)| ≤ 1, we get |f(x)-0| = |x sin(1/x)| ≤ |x| < ε. Note how the function tends to a value even though it is not defined at x=0.

For functions of real numbers or integers, we can also define the limit as x tends to infinity:

definition limx→∞ f(x) = c if, for every number ε > 0, there exists a number X, such that |f(x)-c| < ε whenever x > X.

Since a sequence can be considered formally as a function from the naturals, the above also gives a definition of limits of sequences. Writing it out in words, a sequence {un} of real numbers tends to a limit c if for each ε>0, there exists an N such that |un-c| < ε for each n > N.

One can of course generalize the concept of limits to more kinds of mathematical objects than numbers:

A sequence {un} of elements from a metric space is said to tend to a limit c if for each ε>0, there exists an N such that d(un, c) < ε for all n > N.

A sequence {un} of elements from a topological space is said to tend to a limit c if for each open set U containing c there is an N such that un is in U for each n > N. Limits are unique iff the space is Hausdorff, which is the case for example if it is metrizable. Instead of sequences, i.e. functions from the the natural numbers, we can consider functions from some arbitrary directed set (A,≤). Such a generalized sequence is called a net. Every sequence is of course a net, but unlike sequences, nets can be "uncountably long".

The reals can be considered as a metric space with the metric d(x,y) = |x-y|. Likewise, any metric space can be considered as a topological space whith the open sets being the sets that are open under the metric in question. Then these definitions are all equivalent.

Lim"it (?), n. [From L. limes, limitis: cf. F.limite; -or from E. limit, v. See Limit, v. t.]

1.

That which terminates, circumscribes, restrains, or confines; the bound, border, or edge; the utmost extent; as, the limit of a walk, of a town, of a country; the limits of human knowledge or endeavor.

As eager of the chase, the maid Beyond the forest's verdant limits strayed. Pope.

2.

The space or thing defined by limits.

The archdeacon hath divided it Into three limits very equally. Shak.

3.

That which terminates a period of time; hence, the period itself; the full time or extent.

The dateless limit of thy dear exile. Shak.

The limit of your lives is out. Shak.

4.

A restriction; a check; a curb; a hindrance.

I prithee, give no limits to my tongue. Shak.

5. Logic & Metaph.

A determining feature; a distinguishing characteristic a differentia.

6. Math.

A determinate quantity, to which a variable one continually approaches, and may differ from it by less than any given difference, but to which, under the law of variation, the variable can never become exactly equivalent.

Elastic limit. See under Elastic. -- Prison limits, a definite, extent of space in or around a prison, within which a prisoner has liberty to go and come.

Syn. -- Boundary; border; edge; termination; restriction; bound; confine.

Lim"it (?), v. t. [imp. & p. p. Limited; p. pr. & vb. n. Limiting.] [F. limiter, L. limitare, fr. limes, limitis, limit; prob. akin to limen threshold, E. eliminate; cf. L. limus sidelong.]

To apply a limit to, or set a limit for; to terminate, circumscribe, or restrict, by a limit or limits; as, to limit the acreage of a crop; to limit the issue of paper money; to limit one's ambitions or aspirations; to limit the meaning of a word.

Limiting parallels Astron., those parallels of latitude between which only an occultation of a star or planet by the moon, in a given case, can occur.

Lim"it, v. i.

To beg, or to exercise functions, within a certain limited region; as, a limiting friar.

[Obs.]

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