If the media agency responsible for the authorized production allows material from fans, what is the limit before legal constraints from actors, music, and other considerations, come into play?

If this limit doesn't exist then there is no oblique asymptote in that direction.

If this limit fails to exist then there is no oblique asymptote in that direction, even should the limit defining m exist.

If α is an integer, the limit has to be calculated.

If a main-sequence star is not too massive ( less than approximately 8 solar masses ), it will eventually shed enough mass to form a white dwarf having mass below the Chandrasekhar limit, which will consist of the former core of the star.

This is a Cauchy sequence of rational numbers, but it does not converge towards any rational limit: If the sequence did have a limit x, then necessarily x < sup > 2 </ sup > = 2, yet no rational number has this property.

If the limit exists, then f is differentiable at a.

If the limit exists, meaning that there is a way of choosing a value for Q ( 0 ) that makes the graph of Q a continuous function, then the function f is differentiable at a, and its derivative at a equals Q ( 0 ).

If it can be claimed that it is ethical to limit the internet and other technology to only users who have the means to utilize these software, then there is no argument against the way things are at the moment ; there is no need to complain if all morality is in affect.

Given a trigonometric series f ( x ) with S as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had S ' as its set of zeros, where S ' is the set of limit points of S. If p ( 1 ) is the set of limit points of S, then he could construct a trigonometric series whose zeros are p ( 1 ).

If and then the limit is where is the complete elliptic integral of the first kind

If the limit exists, we say that ƒ is complex-differentiable at the point z < sub > 0 </ sub >.

This can be verified simply by investigating, e. g., a polycrystalline material under a polarizing microscope having the polarizers crossed: If the crystallites are larger than the resolution limit, they will be visible.

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.

If a claims adjuster suspects under-insurance, the condition of average may come into play to limit the insurance company's exposure.

If the time limit is reached and only one competitor has a point, that competitor wins.

If the brightness of the object exceeded the maximum limit of eight times the brightness of Canopus, the spacecraft would search for a new star.

If the offer is for a longer period courts will limit the offer period to 90 days.

If ( x < sub > α </ sub >) is a net in the topological space X, and x is an element of X, we say that the net converges towards x or has limit x and write

If a criminal is on the run, he can be convicted in absence, in order to prevent prescription, or the time limit does not elapse during that time.

If no specific organization plan exists limiting the number of scientists at each salary level, the result is a department top-heavy with high-level, high-salaried personnel ''.

If this be true, the possibility exists that an occlusive lesion of the bronchial arteries might cause widespread degeneration of supportive tissue similar to that seen in generalized emphysema.

If we cannot make explicit choices, how do we know that our set exists?

Statements such as the Banach – Tarski paradox can be rephrased as conditional statements, for example, " If AC holds, the decomposition in the Banach – Tarski paradox exists.

** If the set A is infinite, then there exists an injection from the natural numbers N to A ( see Dedekind infinite ).

If F is an antiderivative of f, and the function f is defined on some interval, then every other antiderivative G of f differs from F by a constant: there exists a number C such that G ( x ) = F ( x ) + C for all x.

The tensor product X ⊗ Y from X and Y is a K-vector space Z with a bilinear function T: X × Y → Z which has the following universal property: If T ′: X × Y → Z ′ is any bilinear function into a K-vector space Z ′, then only one linear function f: Z → Z ′ with exists.

If community exists, both freedom and security may exist as well.

* If the metric space X is compact and an open cover of X is given, then there exists a number such that every subset of X of diameter < δ is contained in some member of the cover.

If a vector field F with zero divergence is defined on a ball in R < sup > 3 </ sup >, then there exists some vector field G on the ball with F = curl ( G ).

If neither A nor B includes the idea of existence, then " some A are B " simply adjoins A to B. Conversely, if A or B do include the idea of existence in the way that " triangle " contains the idea " three angles equal to two right angles ", then " A exists " is automatically true, and we have an ontological proof of A's existence.

If total cash available is less than cash needs, a deficiency exists.

If K is a subset of ker ( f ) then there exists a unique homomorphism h: G / K → H such that f = h φ.

If it exists, the graviton is expected to be massless ( because the gravitational force appears to have unlimited range ) and must be a spin 2 boson.

If R is an integral domain then any two gcd's of a and b must be associate elements, since by definition either one must divide the other ; indeed if a gcd exists, any one of its associates is a gcd as well.

If there exists an isomorphism between two groups, then the groups are called isomorphic.

If God exists in the understanding, we could imagine Him to be greater by existing in reality.

If such a function exists, we say X and Y are homeomorphic.

* If V is a normed vector space with linear subspace U ( not necessarily closed ) and if is continuous and linear, then there exists an extension of φ which is also continuous and linear and which has the same norm as φ ( see Banach space for a discussion of the norm of a linear map ).

* If V is a normed vector space with linear subspace U ( not necessarily closed ) and if z is an element of V not in the closure of U, then there exists a continuous linear map with ψ ( x ) = 0 for all x in U, ψ ( z ) = 1, and || ψ || = 1 / dist ( z, U ).

If an inverse function exists for a given function ƒ, it is unique: it must be the inverse relation.