Limit (mathematics)

In Mathématiques, to seek the limit of a continuation or a function, it is to determine if this continuation or this function approaches a particular value when the variable takes extreme values. In this very intuitive definition, two concepts remain to be defined with precision: concept to approach and that extreme value.

Historically, mathematics initially was interested in the limits of continuations, one sought to know, if, for the great values of the index, the terms of the continuation approached a particular value, i.e., so starting from a certain row, one was as close as one wants this value particular. The concept of proximity is related to a distance which in R is defined by the absolute Value of a difference, but this concept can spread with all metric Espace. Later, the concept extended to the topological Espace S and “close being” then means “to be in a arbitrarily selected Voisinage”.

Then the concept of limit of function, initially attached to the limit of continuation intervened. To seek the limit of a function when the variable approaches has , one sought to determine the limit of the continuation (F (u_n)) for any continuation (u_n) whose limit was has . The complexity of this approach, the multiplicity of the cases, resulted in defining the concept of limit of function independently of that of limit of continuation. To be able to handle the concept of limit and to exploit it without error, it was necessary to define it in a more precise and more formal way. Thus this article presents a formal definition of the limit of a convergent continuation, limit of a function with values in R , the concept of infinite limit, and presents the case of metric space and topological space.

See also, for a more accessible presentation, the article limit in the series Mathematical elementary.

Limit of a succession of real numbers

See also: Limit of continuation

Let us suppose that ( X ₁, X ₂,…) that is to say a continuation of real numbers. One says that this continuation is convergent , so by definition:

there exists a reality L such as

for any reality ε>0 there exists a natural entirety N ₀ (which depends on ε) such as for entire N > N ₀ one has | X _N - L | < ε.

Intuitively, that means that all the terms of the continuation become as close as one wants of a reality L , as soon as N is rather large; the absolute Value | X _N - L | must be interpreted as the “distance” between X _N and L .

It is shown that, for a convergent continuation, reality L of the definition is single. This reality L is called the limit this continuation and one writes:

$$

\ lim_ {N \ to + \ infty} x_n = L

or

$$

\ lim (x_n) = L

All the continuations are not convergent and, if a continuation is not convergent , it is known as nonconvergent or divergent . Some prefer to hold the term diverge with the nonconvergent continuations not limited .

Examples

• the continuation (1/1, 1/2, 1/3, 1/4.) real numbers is convergent of limit 0.

• the continuation (3, 3,3,3,3,…) of limit 3.
• is convergent the continuation (1, -1,1,-1,1,…)
• is nonconvergent the continuation (1, -2,3,-4,5,…) is divergent.
• the continuation (1/2, 1/2 + 1/4, 1/2 + 1/4 + 1/8, 1/2 + 1/4 + 1/8 + 1/16,…) is convergent of limit 1. This continuation is an example of series.
• If has is a real number of absolute Value | has | < 1, then the continuation of general term has ^N has a limit 0.

If has > 0, then the continuation of general term has ^{1/ N} has a limit equalizes to 1.

Properties

Theorem of sequential continuity:

A function F : R → R is continuous in a point L if and only if:

for any real continuation ( X _N ) convergent of limit L , the continuation ( F ( X _N )) is convergent of limit F ( L ).

A under-continuation (or extracted continuation) of the continuation ( X _N ) is a continuation of the form ( X _{has ( N )} ) where the has ( N ) are natural entireties such as for all N one has has ( N ) < has ( N +1). Intuitively, a under-continuation is obtained starting from the initial continuation by omitting certain terms. A continuation is convergent if and only if all its under-continuations are convergent and have even limit.

The operation of passage in extreme cases is linear in the following direction: if ( X _N ) and ( there _N ) are convergent real continuations and that lim X _N = L and lim there _N = P , then the continuation ( X _N + there _N ) is also convergent and has as a limit L + P . If has is a real number, then the continuation ( has X _N ) is convergent of limit Al . Thus, the unit C of all the convergent real continuations is a vector Space real and the operation of passage in extreme cases is a linear form on C with actual values.

If ( X _N ) and ( there _N ) are convergent real continuations of respective limits L and P , then the continuation ( X _N there _N ) is convergent of limit LP . If neither P nor none the terms _N is null there, then the continuation ( X _N / there _N ) is convergent of limit L / P .

Any convergent continuation is a Suite of Cauchy and is thus limited. If ( X _N ) is a succession of realities, limited and increasing (i.e for entire N , X _N ≤ X _{N +1}), then it is necessarily convergent.

Any continuation of Cauchy of real numbers is convergent, or more simply: the whole of realities is complete.

A succession of real numbers is convergent if and only if its lower and higher limits are finished and equal.

Limit of a function in a point

Let us suppose that F : U → R is a function, where U is a subset of the whole of realities. If p is a point of U (Point of accumulation of U ). It is said that F admits a limit (finished) at the point p , if there exists a reality L checking

for any reality ε > 0 there exists a reality δ > 0 such as for all X in U such as | X - p | < δ, one has | F ( X ) - L | < ε.

It is shown that this reality L of the definition is single and it is called limit of F at the point p . It is noted:

$$

\ lim_ {X \ to p} F (X) = L

This is equivalent saying:

for any convergent continuation ( X _N ) in U of limit equalizes with p , the continuation ( F ( X _N )) is convergent of limit L .

Let us notice that the function does not need to be defined in p , but if the function is defined in p then F ( p ) is equal to the limit of F in p and thus the application is continuous in p .

Now let us define the pointed limit:

It is said that F admits a limit pointed (finished) at the point p , if there exists a reality L checking

for any reality ε > 0 there exists a reality δ > 0 such as for all X in U such as 0 < | X - p | < δ, one has | F ( X ) - L | < ε.

The same this number L is then single and one notes:

$$

\ lim_ {X \ to p, X \ neq p} F (X) = L

Occasionally, it can be useful to approach the point p that on only one side.

It is said that F admits a limit on the right (finished) at the point p , if there exists a reality L checking

for any reality ε > 0 there exists a reality δ > 0 such as for all X in U such as 0 < X - p < δ, one has | F ( X ) - L | < ε.

This number L is then single and it is noted:

$$

\ lim_ {X \ to p+} F (X) = L

The limits on the left are obtained by replacing X - p in the last definition by p - X .

It is possible also to consider limits where p or L is equal to more or less the infinite one. It is said that F ( X ) tends towards more the infinite one (+∞) when X tends towards p so by definition

for any reality R > 0, there exists a reality δ > 0 such as for all X such as | X - p | < δ one has F ( X ) > R .

It is said that limit of F ( X ) when X tends towards more the infinite one is equal to L if

for any reality ε > 0 there exists a reality S > 0 such as for all X > S , one has | F ( X ) - L | < ε.

Lastly, it is said that the limit of F ( X ) is equal to more the infinite one when X tends towards more the infinite one, if

for any reality R > 0 there exists a reality S > 0 such as for all X such as X > S , one has F ( X ) > R .

The definitions for less the infinite one are similar.

By replacing ε by S like previously, one can also define the infinite limits on only one side (on the right or on the left).

Examples

• the limit of $x\; \backslash \; mapsto\; \backslash \; frac\; \{1\}\; \{X\}$ when X tends towards the infinite one is equal to 0.
  

$\backslash \; lim\_\; \{X\; \backslash \; to\; \backslash \; infty\}\; \backslash \; frac\; \{1\}\; \{X\}\; =0$

• the limits on the right and on the left of $x\; \backslash \; mapsto\; \backslash \; frac\; \{1\}\; \{X\}$ when X tends towards 0 does not exist. The limit of $x\; \backslash \; mapsto\; \backslash \; frac\; \{1\}\; \{X\}$ when X tends towards 0 by higher values is +∞.
  

$\backslash \; lim\_\; \{X\; \backslash \; to\; 0+\}\; \backslash \; frac\; \{1\}\; \{X\}\; =+\; \backslash \; infty\; \backslash \; qquad\; \backslash \; lim\_\; \{X\; \backslash \; to\; 0\}\; \backslash \; frac\; \{1\}\; \{X\}\; =\; \backslash \; infty$

• the limit of $x\; \backslash \; mapsto\; x^2$ when X tends towards 3 is equal to 9. (In this case the function is definite and continuous in this point, and the value of the function is equal to the limit.)
  

$\backslash \; lim\_\; \{X\; \backslash \; to\; 3\}\; x^2=9$

• the limit of $x\; \backslash \; mapsto\; x^x$ when X tends towards 0 is equal to 1.
  

$\backslash \; lim\_\; \{X\; \backslash \; to\; 0\}\; x^x=1$

• the limit of $x\; \backslash \; mapsto\; \backslash \; frac\; \{(a+\; X)\; ^2-a^2\}\; \{X\}$ when X tends towards 0 is equal to 2 has .
  

$\backslash \; lim\_\; \{X\; \backslash \; to\; 0\}\; \backslash \; frac\; \{(a+\; X)\; ^2-a^2\}\; \{X\}\; =2a$

• the limit on the right of $x\; \backslash \; mapsto\; \backslash \; frac\; \{\backslash \; sqrt\; \{x^2\}\}\; \{X\}$ when X tends towards 0 by higher values is equal to 1; the limit on the left is equal to -1.
  

$\backslash \; lim\_\; \{X\; \backslash \; to\; 0+\}\; \backslash \; frac\; \{\backslash \; sqrt\; \{x^2\}\}\; \{X\}\; =1$$\backslash \; lim\_\; \{X\; \backslash \; to\; 0\}\; \backslash \; frac\; \{\backslash \; sqrt\; \{x^2\}\}\; \{X\}\; =-1$

• limit of $x\; \backslash \; mapsto\; X.\; \backslash \; sin\; \backslash \; frac\; \{1\}\; \{X\}$ when X tends towards more the infinite one is equal to 1.
  

$\backslash \; lim\_\; \{X\; \backslash \; to\; +\; \backslash \; infty\}\; X.\; \backslash \; sin\; \backslash \; frac\; \{1\}\; \{X\}\; =1$

• limit of $x\; \backslash \; mapsto\; \backslash \; frac\; \{\backslash \; cos\; (X)\; -\; 1\}\; \{X\}$ when X tends towards 0 is equal to 0.
  

$\backslash \; lim\_\; \{X\; \backslash \; to\; 0\}\; \backslash \; frac\; \{\backslash \; cos\; (X)\; -\; 1\}\; \{X\}\; =0$

Properties

See also: Operations on the limits, elementary Theorems on the limits

Pointed limit of F ( X ) when X tends towards p exists if and only if the limits on the right and on the left in p exist and are equal.

If p is a point of U , then limit of F ( X ) when X tends towards p exists if and only if the limits on the right and on the left in p exist and are equal to F ( p ), if and only if pointed limit of F ( X ) when X tends towards p exists and is equal to F ( p ) and if and only if F is continuous in p .

If p does not belong U then the limit of F ( X ) when X tends towards p exists if and only if the limits on the right and on the left in p exist and are equal.

The passage in extreme cases of the functions is compatible with the algebraic operations:

If

$$

\ lim_ {X \ to p} f_1 (X) = L_1 and

$$

\ lim_ {X \ to p} f_2 (X) = L_2

then

$$

\ lim_ {X \ to p} (f_1 (X) + f_2 (X)) = L_1 + L_2

and

$$

\ lim_ {X \ to p} (f_1 (X) \ times f_2 (X)) = L_1 \ times L_2

and

$$

\ lim_ {X \ to p} \ left (\ frac {f_1 (X)} {f_2 (X)}\ right) = \ frac {L_1} {L_2}

(The last property supposes that F ₂ is not cancelled in a vicinity of p and that L ₂ is not null).

These properties are also valid for the limits on the right and on the left, the case p = ±∞, and also for the infinite limits by using the following rules:

• Q + ∞ = ∞ for Q ≠ - ∞
• Q × ∞ = ∞ if Q > 0
• Q × ∞ = - ∞ if Q < 0
• Q /∞ = 0 if Q ≠ ± ∞

(see the completed real Right ).

Let us notice that there is no general rule for the case Q /0; that depends on the way in which one approaches 0. Certain cases such as for example 0/0, 0×∞ ∞-∞ or ∞/∞, are not covered either by these rules but the limits can be in general obtained by the Règle of the Hospital.

Indetermination

There exist certain forms of limit where it is not possible to conclude directly by using Opérations on the limits, they are the forms known as unspecified.

Indetermination of form 0/0 when the result obtained gives 0/0

Indetermination of the form ∞/∞ when the result obtained gives ∞/∞

Indetermination of the form ∞ - ∞ when the result obtained gives ∞ - ∞

Indétermination of the form 0 × ∞ which is reduced to the first two cases by noticing that a multiplication by 0 is equivalent to a division by the infinite one, or that a multiplication by the infinite one is equivalent to a division by 0

Indetermination of the form 0⁰ which is reduced to the preceding case by noticing that a^b can be written e^{b×ln (a)} and that the limit of b×ln (a) is then of the form 0 × ∞

One can use the Règle of the Hospital to raise an indetermination during a calculation of limit.

Metric spaces

The real numbers form a metric Espace if we use the function outdistances definite by the absolute value: D ( X , there ) = | X - there |. It is the same of the complex numbers with the module. Moreover, the Euclidean Space R ^N form a metric space with the Euclidean distance. Here some examples justifying a generalization of the definitions of limit given previously.

If ( X _N ) is a continuation in a metric space ( M , D ), then it is said that it following a limit L so by definition for any reality ε>0 there exists a natural entirety N ₀ such as for entire N > N ₀ one has D ( X _N , L ) < ε.

If metric space ( M , D ) is complete (what is the case for the whole of the real numbers or complexes and Euclidean space, and very other Espace of Banach, then one can establish the convergence of a continuation of M by showing that it is a Suite of Cauchy. The advantage of this approach is to be able to show that the continuation is convergent without necessarily knowing the limit in advance.

If M is a vector Space normalized real or complex, then the operation of passage in extreme cases is linear, as we explained above in the case of the continuations of real numbers.

Now let us suppose that F : M → NR is an application between two metric spaces, and that p is an element of M and L an element of NR . It is said that limit of F ( X ) when X tends towards p is equal to L and one writes:

$$

\ lim_ {X \ to p} F (X) = L

so by definition:

for any reality ε > 0 there exists a reality δ > 0 such as for all X in M such as D ( X , p ) < δ, one has D ( F ( X ), L ) < ε.

What is equivalent to

for any convergent continuation ( X _N ) of M such as the limit is equal to p , the continuation ( F ( X _N )) is convergent of limit L .

A function F is continuous in p if and only if limit of F ( X ) when X tends towards p exists and is equal to F ( p ). In an equivalent way, F transforms any convergent continuation of M of limit p into a convergent continuation of NR of limit F ( p ).

Again, if NR is a normalized vector space, then the operation of passage in extreme cases is linear in the following direction: if limit of F ( X ) when X tends towards p is equal to L and limit of G ( X ) when X tends towards p is equal to P , then limit of F ( X ) + G ( X ) when X tends towards p is equal to L + P . If has is a scalar of the basic body, then limit of af ( X ) when X tends towards p is equal to Al .

If NR is equal to R , then we can define limit infinite; if M is equal to R , then we can define limit on the right and on the left in a way similar to the preceding definitions.

Examples

• If Z is a complex number of module | Z | < 1, then the continuation ( Z , Z ², Z ³,…) complex numbers converges and has as a limit 0. Geometrically, these numbers approach the origin while following a “spiral logarithmic curve”.

• In metric space C of all the continuous functions definite on the interval, provided with the distance from uniform convergence, any element can be written like limit of a succession of polynomial functions. It is what affirms the Théorème of Stone-Weierstrass.

Generalizations for topological spaces

All the concepts of limit above can be unified and generalized with a topological Espace arbitrary by introducing the filter S and their limit.

Nevertheless the definition of limit for the particular case of the continuations (indexed by $\backslash \; mathbb\; \{NR\}$) is relatively simple:

• a continuation $(u\_n)\; \_\; \{N\; \backslash \; in\; \{\backslash \; mathbb\; NR\}\}$ converges towards a limit U if and only so for all Voisinage V of U, it exists a natural entirety NR such as $\backslash \; forall\; N\; \backslash \; geq\; NR,\; u\_n\; \backslash \; in\; V$.

One can give an equivalent definition using the bases of vicinities:

• being given $\backslash \; mathcal\; \{W\}$, a base of vicinities of U , a continuation $(u\_n)\; \_\; \{N\; \backslash \; in\; \{\backslash \; mathbb\; NR\}\}$ converges towards U if and only so for all $V\; \backslash \; in\; \backslash \; mathcal\; \{W\}$, it exists a natural entirety NR such as $\backslash \; forall\; N\; \backslash \; geq\; NR,\; u\_n\; \backslash \; in\; V$.

This definition can also be reformulated in term of open:

• a continuation $(u\_n)\; \_\; \{N\; \backslash \; in\; \{\backslash \; mathbb\; NR\}\}$ converges towards a limit U if and only so for all open O container U, it exists a natural entirety NR such as $\backslash \; forall\; N\; \backslash \; geq\; NR,\; u\_n\; \backslash \; in\; O$.

Related articles

• Operations on the elementary limits
• Theorems on the limits
• Theorem of the gendarmes
• Rule of the Hospital

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