It is possible that a diagram does not have a limit at all.

However, if a diagram does have a limit then this limit is essentially unique: it is unique up to a unique isomorphism.

Any collection of objects and morphisms defines a ( possibly large ) directed graph G. If we let J be the free category generated by G, there is a universal diagram F: J → C whose image contains G. The limit ( or colimit ) of this diagram is the same as the limit ( or colimit ) of the original collection of objects and morphisms.

In the following we will consider the limit ( L, φ ) of a diagram F: J → C.

A cone to the empty diagram is essentially just an object of C. The limit of F is any object that is uniquely factored through by every other object.

A special case of a product is when the diagram F is a constant functor to an object X of C. The limit of this diagram is called the J < sup > th </ sup > power of X and denoted X < sup > J </ sup >.

If J is a category with two objects and two parallel morphisms from object 1 to object 2 then a diagram of type J is a pair of parallel morphisms in C. The limit L of such a diagram is called an equalizer of those morphisms.

A morphism f: Y → X is a limit of the diagram X if and only if f is an isomorphism.

More generally, if J is any category with an initial object i, then any diagram of type J has a limit, namely any object isomorphic to F ( i ).

A given diagram F: J → C may or may not have a limit ( or colimit ) in C. Indeed, there may not even be a cone to F, let alone a universal cone.

A category C is said to have limits of type J if every diagram of type J has a limit in C. Specifically, a category C is said to

In this case, the limit of a diagram F: J → C can be constructed as the equalizer of the two morphisms

In other words, if every diagram of type J has a limit in C ( for J small ) there exists a limit functor

which assigns each diagram its limit and each natural transformation η: F → G the unique morphism lim η: lim F → lim G commuting with the corresponding universal cones.

If a diagram F: J → C has a limit in C, denoted by lim F, there is a canonical isomorphism

Identifying the limit of Hom ( F –, N ) with the set Cocone ( F, N ), this relationship can be used to define the colimit of the diagram F as a representation of the functor Cocone ( F, –).

A functor G: C → D is said to lift limits for a diagram F: J → C if whenever ( L, φ ) is a limit of GF there exists a limit ( L ′, φ ′) of F such that G ( L ′, φ ′)

More rigorously, the divergence of a vector field F at a point p is defined as the limit of the net flow of F across the smooth boundary of a three dimensional region V divided by the volume of V as V shrinks to p. Formally,

The warm summer of 1999 caused lake temperatures to come close to the 85 ° F ( 29 ° C ) limit necessary to keep the plants cool.

* The Monster of Compression benchmark by N. F. Antonio tests compression on 1Gb of public data with a 40 minute time limit.

This group is the inverse limit of the finite groups Gal ( F / K ), where F ranges over all intermediate fields such that F / K is a finite Galois extension.

For the limit process, we use the restriction homomorphisms Gal ( F < sub > 1 </ sub >/ K ) → Gal ( F < sub > 2 </ sub >/ K ), where F < sub > 2 </ sub > ⊆ F < sub > 1 </ sub >.

G < sub > 3 </ sub >( F λ ), for all limit λ ≠ 0.

Given a functor F: J → C ( thought of as an object in C < sup > J </ sup >), the limit of F, if it exists, is nothing but a terminal morphism from Δ to F. Dually, the colimit of F is an initial morphism from F to Δ.

Let φ ( ξ, η, ζ ) be an arbitrary function of three independent variables, and let the spherical wave form F be a delta-function: that is, let F be a weak limit of continuous functions whose integral is unity, but whose support ( the region where the function is non-zero ) shrinks to the origin.

For this reason one often speaks of the limit of F.

Limit functor: For a fixed index category J, if every functor J → C has a limit ( for instance if C is complete ), then the limit functor C < sup > J </ sup >→ C assigns to each functor its limit.

In 1993, R. D. Stevenson and R. J. Wassersug published an article calculating the upper limit to an animal's power output.

* An example of a non-unitary associative algebra is given by the set of all functions f: R → R whose limit as x nears infinity is zero.

Divide by Δs and take the limit as Δs → 0 to obtain

If this limit exists, then it may be computed by taking the limit as h → 0 along the real axis or imaginary axis ; in either case it should give the same result.

The Lorentz transformation is equivalent to the Galilean transformation in the limit c < sub > 0 </ sub > → ∞ ( a hypothetical case ) or v → 0 ( low speeds ).

The inverse limit, A, comes equipped with natural projections π < sub > i </ sub >: A → A < sub > i </ sub > which pick out the ith component of the direct product for each i in I.

The inverse limit of this system is an object X in C together with morphisms π < sub > i </ sub >: X → X < sub > i </ sub > ( called projections ) satisfying π < sub > i </ sub >

If it does, however, it is unique in a strong sense: given any other inverse limit X ′ there exists a unique isomorphism X ′ → X commuting with the projection maps.

An inverse system is then just a contravariant functor I → C. And the inverse limit functor

* Formally, when working over the reals, as here, this is accomplished by considering the limit as ε → 0 ; but the " infinitesimal " language generalizes directly to Lie groups over general rings.

It is the inverse limit of the finite groups Z / p < sup > n </ sup > Z where n ranges over all natural numbers and the natural maps Z / p < sup > n </ sup > Z → Z / p < sup > m </ sup > Z ( n ≥ m ) are used for the limit process.

The Dirac delta function as the limit ( in the sense of distribution ( mathematics ) | distributions ) of the sequence of zero-centered normal distribution s as a → 0

The fact that there can not be any limit points of the set except in closed intervals follows from the argument used in Lemma 1, namely, that near any tangent point in the C-plane the curves C and Af are analytic, and therefore the difference between them must be a monotone function in some neighborhood on either side of the tangent point.

This extra limit, however, can be a rather large stumbling block for most C programmers, who are used to being able to manipulate their pointers directly with arithmetic.

Gelatin gels exist over only a small temperature range, the upper limit being the melting point of the gel, which depends on gelatin grade and concentration ( but is typically less than 35 ° C ) and the lower limit the freezing point at which ice crystallizes.

For an abelian category C, the inverse limit functor

More generally, if C is an arbitrary abelian category that has enough injectives, then so does C < sup > I </ sup >, and the right derived functors of the inverse limit functor can thus be defined.

First, the elements C, N and O are only created after at least one cycle of star birth / death: this is a limit to the earliest time life could have arisen.

The southern limit is more variable, depending on rainfall ; taiga may be replaced by forest steppe south of the 15 ° C July isotherm where rainfall is very low, but more typically extends south to the 18 ° C July isotherm, and locally where rainfall is higher ( notably in eastern Siberia and adjacent Outer Manchuria ) south to the 20 ° C July isotherm.