Limit of a function - wikipedia Limit of a function Jump to : navigation , search x sin  x x ( \ displaystyle ( \ frac ( \ sin x ) ( x ) ) ) 0.841471 ... 0.1 0.998334 ... 0.01 0.999983 ... Although the function ( sin x ) / x is not defined at zero , as x becomes closer and closer to zero , ( sin x ) / x becomes arbitrarily close to 1 . In other words , the limit of ( sin x ) / x as x approaches zero equals 1 . Part of a series of articles about Calculus Fundamental theorem Limits of functions Continuity Mean value theorem Rolle 's theorem Differential ( show ) Definitions Derivative ( generalizations ) Differential infinitesimal of a function total Concepts Differentiation notation Second derivative Third derivative Change of variables Implicit differentiation Related rates Taylor 's theorem Rules and identities Sum Product Chain Power Quotient Inverse General Leibniz Faà di Bruno 's formula Integral ( show ) Lists of integrals Definitions Antiderivative Integral ( improper ) Riemann integral Lebesgue integration Contour integration Integration by Parts Discs Cylindrical shells Substitution ( trigonometric ) Partial fractions Order Reduction formulae Series ( show ) Geometric ( arithmetico - geometric ) Harmonic Alternating Power Binomial Taylor Convergence tests Summand limit ( term test ) Ratio Root Integral Direct comparison Limit comparison Alternating series Cauchy condensation Dirichlet Abel Vector ( show ) Gradient Divergence Curl Laplacian Directional derivative Identities Theorems Divergence Gradient Green 's Kelvin -- Stokes Stokes Multivariable ( show ) Formalisms Matrix Tensor Exterior Geometric Definitions Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian Hessian Specialized ( show ) Fractional Malliavin Stochastic Variations Glossary of calculus ( show ) Glossary of calculus In mathematics , the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input . Formal definitions , first devised in the early 19th century , are given below . Informally , a function f assigns an output f ( x ) to every input x . We say the function has a limit L at an input p : this means f ( x ) gets closer and closer to L as x moves closer and closer to p . More specifically , when f is applied to any input sufficiently close to p , the output value is forced arbitrarily close to L. On the other hand , if some inputs very close to p are taken to outputs that stay a fixed distance apart , we say the limit does not exist . The notion of a limit has many applications in modern calculus . In particular , the many definitions of continuity employ the limit : roughly , a function is continuous if all of its limits agree with the values of the function . It also appears in the definition of the derivative : in the calculus of one variable , this is the limiting value of the slope of secant lines to the graph of a function . Contents ( hide ) 1 History 2 Motivation 3 Functions of a single variable 3.1 Existence and one - sided limits 3.2 More general subsets 3.3 Deleted versus non-deleted limits 3.4 Examples 3.4. 1 Non-existence of one - sided limit ( s ) 3.4. 2 Non-equality of one - sided limits 3.4. 3 Limits at only one point 3.4. 4 Limits at countably many points 4 Functions on metric spaces 5 Functions on topological spaces 6 Limits involving infinity 6.1 Limits at infinity 6.2 Infinite limits 6.3 Alternative notation 6.4 Limits at infinity for rational functions 7 Functions of more than one variable 8 Sequential limits 9 Other characterizations 9.1 In terms of sequences 9.2 In non-standard calculus 9.3 In terms of nearness 10 Relationship to continuity 11 Properties 11.1 Limits of compositions of functions 11.2 Limits of special interest 11.2. 1 Rational functions 11.2. 2 Trigonometric functions 11.2. 3 Exponential functions 11.2. 4 Logarithmic functions 11.3 L'Hôpital's rule 11.4 Summations and integrals 12 See also 13 Notes 14 References History ( edit ) Although implicit in the development of calculus of the 17th and 18th centuries , the modern idea of the limit of a function goes back to Bolzano who , in 1817 , introduced the basics of the epsilon - delta technique to define continuous functions . However , his work was not known during his lifetime ( Felscher 2000 ) . Cauchy discussed variable quantities , infinitesimals , and limits and defined continuity of y = f ( x ) ( \ displaystyle y = f ( x ) ) by saying that an infinitesimal change in x necessarily produces an infinitesimal change in y in his 1821 book Cours d'analyse , while ( Grabiner 1983 ) claims that he only gave a verbal definition . Weierstrass first introduced the epsilon - delta definition of limit in the form it is usually written today . He also introduced the notations lim and lim ( Burton 1997 ) . The modern notation of placing the arrow below the limit symbol is due to Hardy in his book A Course of Pure Mathematics in 1908 ( Miller 2004 ) . Motivation ( edit ) Imagine a person walking over a landscape represented by the graph of y = f ( x ) . Her horizontal position is measured by the value of x , much like the position given by a map of the land or by a global positioning system . Her altitude is given by the coordinate y . She is walking towards the horizontal position given by x = p . As she gets closer and closer to it , she notices that her altitude approaches L. If asked about the altitude of x = p , she would then answer L . What , then , does it mean to say that her altitude approaches L ? It means that her altitude gets nearer and nearer to L except for a possible small error in accuracy . For example , suppose we set a particular accuracy goal for our traveler : she must get within ten meters of L. She reports back that indeed she can get within ten meters of L , since she notes that when she is within fifty horizontal meters of p , her altitude is always ten meters or less from L . The accuracy goal is then changed : can she get within one vertical meter ? Yes . If she is anywhere within seven horizontal meters of p , then her altitude always remains within one meter from the target L. In summary , to say that the traveler 's altitude approaches L as her horizontal position approaches p means that for every target accuracy goal , however small it may be , there is some neighborhood of p whose altitude fulfills that accuracy goal . The initial informal statement can now be explicated : The limit of a function f ( x ) as x approaches p is a number L with the following property : given any target distance from L , there is a distance from p within which the values of f ( x ) remain within the target distance . This explicit statement is quite close to the formal definition of the limit of a function with values in a topological space . To say that lim x → p f ( x ) = L , ( \ displaystyle \ lim _ ( x \ to p ) f ( x ) = L , \ , ) means that ƒ ( x ) can be made as close as desired to L by making x close enough , but not equal , to p . The following definitions ( known as ( ε , δ ) - definitions ) are the generally accepted ones for the limit of a function in various contexts . Functions of a single variable ( edit ) Suppose f : R → R is defined on the real line and p , L ∈ R. It is said the limit of f , as x approaches p , is L and written lim x → p f ( x ) = L , ( \ displaystyle \ lim _ ( x \ to p ) f ( x ) = L , \ ) if the following property holds : For every real ε > 0 , there exists a real δ > 0 such that for all real x , 0 < x − p < δ implies f ( x ) − L < ε . The value of the limit does not depend on the value of f ( p ) , nor even that p be in the domain of f . A more general definition applies for functions defined on subsets of the real line . Let ( a , b ) be an open interval in R , and p a point of ( a , b ) . Let f be a real - valued function defined on all of ( a , b ) except possibly at p itself . It is then said that the limit of f , as x approaches p , is L if , for every real ε > 0 , there exists a real δ > 0 such that 0 < x − p < δ and x ∈ ( a , b ) implies f ( x ) − L < ε . Here again the limit does not depend on f ( p ) being well - defined . The letters ε and δ can be understood as `` error '' and `` distance '' , and in fact Cauchy used ε as an abbreviation for `` error '' in some of his work ( Grabiner 1983 ) , though in his definition of continuity he used an infinitesimal α ( \ displaystyle \ alpha ) rather than either ε or δ ( see Cours d'Analyse ) . In these terms , the error ( ε ) in the measurement of the value at the limit can be made as small as desired by reducing the distance ( δ ) to the limit point . As discussed below this definition also works for functions in a more general context . The idea that δ and ε represent distances helps suggest these generalizations . Existence and one - sided limits ( edit ) Main article : One - sided limit The limit as : x → x ≠ x → x . Therefore , the limit as x → x does not exist . Alternatively x may approach p from above ( right ) or below ( left ) , in which case the limits may be written as lim x → p + f ( x ) = L ( \ displaystyle \ lim _ ( x \ to p ^ ( + ) ) f ( x ) = L ) or lim x → p − f ( x ) = L ( \ displaystyle \ lim _ ( x \ to p ^ ( - ) ) f ( x ) = L ) respectively . If these limits exist at p and are equal there , then this can be referred to as the limit of f ( x ) at p . If the one - sided limits exist at p , but are unequal , there is no limit at p ( the limit at p does not exist ) . If either one - sided limit does not exist at p , the limit at p does not exist . A formal definition is as follows . The limit of f ( x ) as x approaches p from above is L if , for every ε > 0 , there exists a δ > 0 such that f ( x ) − L < ε whenever 0 < x − p < δ . The limit of f ( x ) as x approaches p from below is L if , for every ε > 0 , there exists a δ > 0 such that f ( x ) − L < ε whenever 0 < p − x < δ . If the limit does not exist then the oscillation of f at p is non-zero . More general subsets ( edit ) Apart from open intervals , limits can be defined for functions on arbitrary subsets of R , as follows . Let f be a real - valued function defined on a subset S of the real line . Let p be a limit point of S -- that is , p is the limit of some sequence of elements of S distinct from p . The limit of f , as x approaches p from values in S , is L if , for every ε > 0 , there exists a δ > 0 such that 0 < x − p < δ and x ∈ S implies f ( x ) − L < ε . This limit is often written L = lim x → p x ∈ S f ( x ) . ( \ displaystyle L = ( \ underset ( x \ in S ) ( \ lim _ ( x \ to p ) ) ) f ( x ) . ) The condition that f be defined on S is that S be a subset of the domain of f . This generalization includes as special cases limits on an interval , as well as left - handed limits of real - valued functions ( e.g. , by taking S to be an open interval of the form ( − ∞ , a ) ( \ displaystyle ( - \ infty , a ) ) ) , and right - handed limits ( e.g. , by taking S to be an open interval of the form ( a , ∞ ) ( \ displaystyle ( a , \ infty ) ) ) . It also extends the notion of one - sided limits to the included endpoints of ( half - ) closed intervals , so the Square root function f ( x ) = √ x can have limit 0 as x approaches 0 from above . Deleted versus non-deleted limits ( edit ) The definition of limit given here does not depend on how ( or whether ) f is defined at p . Bartle ( 1967 ) , refers to this as a deleted limit , because it excludes the value of f at p . The corresponding non-deleted limit does depend on the value of f at p , if p is in the domain of f : A number L is the non-deleted limit of f as x approaches p if , for every ε > 0 , there exists a δ > 0 such that x − p < δ and x ∈ Dm ( f ) implies f ( x ) − L < ε . The definition is the same , except that the neighborhood x − p < δ now includes the point p , in contrast to the ( deleted ) neighborhood 0 < x − p < δ . This makes the definition of a non-deleted limit less general . One of the advantages of working with non-deleted limits is that they allow to state the theorem about limits of compositions without any constraints on the functions ( other than the existence of their non-deleted limits ) ( Hubbard ( 2015 ) ) . Bartle ( 1967 ) notes that although by `` limit '' some authors do mean this non-deleted limit , deleted limits are the most popular . For example , Apostol ( 1974 ) , Courant ( 1924 ) , Hardy ( 1921 ) , Rudin ( 1964 ) , Whittaker & Watson ( 1902 ) all by `` limit '' mean the deleted version . Examples ( edit ) Non-existence of one - sided limit ( S ) ( edit ) The function without a limit , at an essential discontinuity The function f ( x ) = ( sin  5 x − 1 for x < 1 0 for x = 1 0.1 x − 1 for x > 1 ( \ displaystyle f ( x ) = ( \ begin ( cases ) \ sin ( \ frac ( 5 ) ( x-1 ) ) & ( \ text ( for ) ) x < 1 \ \ 0& ( \ text ( for ) ) x = 1 \ \ ( \ frac ( 0.1 ) ( x-1 ) ) & ( \ text ( for ) ) x > 1 \ end ( cases ) ) ) has no limit at x 0 = 1 ( \ displaystyle x_ ( 0 ) = 1 ) ( the left - hand limit does not exist due to the oscillatory nature of the sine function , and the right - hand limit does not exist due to the asymptotic behaviour of the reciprocal function ) , but has a limit at every other x-coordinate . The function f ( x ) = ( 1 x rational 0 x irrational ( \ displaystyle f ( x ) = ( \ begin ( cases ) 1&x ( \ text ( rational ) ) \ \ 0&x ( \ text ( irrational ) ) \ end ( cases ) ) ) ( the Dirichlet function ) has no limit at any x-coordinate . Non-equality of one - sided limits ( edit ) The function f ( x ) = ( 1 for x < 0 2 for x ≥ 0 ( \ displaystyle f ( x ) = ( \ begin ( cases ) 1& ( \ text ( for ) ) x < 0 \ \ 2& ( \ text ( for ) ) x \ geq 0 \ end ( cases ) ) ) has a limit at every non-zero x-coordinate ( the limit equals 1 for negative x and equals 2 for positive x ) . The limit at x = 0 does not exist ( the left - hand limit equals 1 , whereas the right - hand limit equals 2 ) . Limits at only one point ( edit ) The functions f ( x ) = ( x x rational 0 x irrational ( \ displaystyle f ( x ) = ( \ begin ( cases ) x&x ( \ text ( rational ) ) \ \ 0&x ( \ text ( irrational ) ) \ end ( cases ) ) ) and f ( x ) = ( x x rational 0 x irrational ( \ displaystyle f ( x ) = ( \ begin ( cases ) x &x ( \ text ( rational ) ) \ \ 0&x ( \ text ( irrational ) ) \ end ( cases ) ) ) both have a limit at x = 0 and it equals 0 . Limits at countably many points ( edit ) The function f ( x ) = ( sin  x x irrational 1 x rational ( \ displaystyle f ( x ) = ( \ begin ( cases ) \ sin x&x ( \ text ( irrational ) ) \ \ 1&x ( \ text ( rational ) ) \ end ( cases ) ) ) has a limit at any x-coordinate of the form π 2 + 2 n π ( \ displaystyle ( \ frac ( \ pi ) ( 2 ) ) + 2n \ pi ) , where n is any integer . Functions on metric spaces ( edit ) Suppose M and N are subsets of metric spaces A and B , respectively , and f : M → N is defined between M and N , with x ∈ M , p a limit point of M and L ∈ N. It is said that the limit of f as x approaches p is L and write lim x → p f ( x ) = L ( \ displaystyle \ lim _ ( x \ to p ) f ( x ) = L ) if the following property holds : For every ε > 0 , there exists a δ > 0 such that d ( f ( x ) , L ) < ε whenever 0 < d ( x , p ) < δ . Again , note that p need not be in the domain of f , nor does L need to be in the range of f , and even if f ( p ) is defined it need not be equal to L . An alternative definition using the concept of neighbourhood is as follows : lim x → p f ( x ) = L ( \ displaystyle \ lim _ ( x \ to p ) f ( x ) = L ) if , for every neighbourhood V of L in B , there exists a neighbourhood U of p in A such that f ( U ∩ M − ( p ) ) ⊆ V . Functions on topological spaces ( edit ) Suppose X , Y are topological spaces with Y a Hausdorff space . Let p be a limit point of Ω ⊆ X , and L ∈ Y . For a function f : Ω → Y , it is said that the limit of f as x approaches p is L ( i.e. , f ( x ) → L as x → p ) and written lim x → p f ( x ) = L ( \ displaystyle \ lim _ ( x \ to p ) f ( x ) = L ) if the following property holds : For every open neighborhood V of L , there exists an open neighborhood U of p such that f ( U ∩ Ω − ( p ) ) ⊆ V . This last part of the definition can also be phrased `` there exists an open punctured neighbourhood U of p such that f ( U ∩ Ω ) ⊆ V `` . Note that the domain of f does not need to contain p . If it does , then the value of f at p is irrelevant to the definition of the limit . In particular , if the domain of f is X − ( p ) ( or all of X ) , then the limit of f as x → p exists and is equal to L if , for all subsets Ω of X with limit point p , the limit of the restriction of f to Ω exists and is equal to L. Sometimes this criterion is used to establish the non-existence of the two - sided limit of a function on R by showing that the one - sided limits either fail to exist or do not agree . Such a view is fundamental in the field of general topology , where limits and continuity at a point are defined in terms of special families of subsets , called filters , or generalized sequences known as nets . Alternatively , the requirement that Y be a Hausdorff space can be relaxed to the assumption that Y be a general topological space , but then the limit of a function may not be unique . In particular , one can no longer talk about the limit of a function at a point , but rather a limit or the set of limits at a point . A function is continuous in a limit point p of and in its domain if and only if f ( p ) is the ( or , in the general case , a ) limit of f ( x ) as x tends to p . Limits involving infinity ( edit ) Limits at infinity ( edit ) The limit of this function at infinity exists . For f ( x ) a real function , the limit of f as x approaches infinity is L , denoted lim x → ∞ f ( x ) = L , ( \ displaystyle \ lim _ ( x \ to \ infty ) f ( x ) = L , ) means that for all ε > 0 ( \ displaystyle \ varepsilon > 0 ) , there exists c such that f ( x ) − L < ε ( \ displaystyle f ( x ) - L < \ varepsilon ) whenever x > c . Or , symbolically : ∀ ε > 0 ∃ c ∀ x > c : f ( x ) − L < ε ( \ displaystyle \ forall \ varepsilon > 0 \ ; \ exists c \ ; \ forall x > c : \ ; f ( x ) - L < \ varepsilon ) . Similarly , the limit of f as x approaches negative infinity is L , denoted lim x → − ∞ f ( x ) = L , ( \ displaystyle \ lim _ ( x \ to - \ infty ) f ( x ) = L , ) means that for all ε > 0 ( \ displaystyle \ varepsilon > 0 ) there exists c such that f ( x ) − L < ε ( \ displaystyle f ( x ) - L < \ varepsilon ) whenever x < c . Or , symbolically : ∀ ε > 0 ∃ c ∀ x < c : f ( x ) − L < ε ( \ displaystyle \ forall \ varepsilon > 0 \ ; \ exists c \ ; \ forall x < c : \ ; f ( x ) - L < \ varepsilon ) . For example lim x → − ∞ e x = 0 . ( \ displaystyle \ lim _ ( x \ to - \ infty ) e ^ ( x ) = 0. \ , ) Infinite limits ( edit ) Limits can also have infinite values . When infinities are not considered legitimate values , which is standard ( but see below ) , a formalist will insist upon various circumlocutions . For example , rather than say that a limit is infinity , the proper thing is to say that the function `` diverges '' or `` grows without bound '' . In particular , the following informal example of how to pronounce the notation is arguably inappropriate in the classroom ( or any other formal setting ) . In any case , for example the limit of f as x approaches a is infinity , denoted lim x → a f ( x ) = ∞ , ( \ displaystyle \ lim _ ( x \ to a ) f ( x ) = \ infty , \ , ) means that for all ε > 0 ( \ displaystyle \ varepsilon > 0 ) there exists δ > 0 ( \ displaystyle \ delta > 0 ) such that f ( x ) > ε ( \ displaystyle f ( x ) > \ varepsilon ) whenever x − a < δ ( \ displaystyle x-a < \ delta ) . These ideas can be combined in a natural way to produce definitions for different combinations , such as lim x → ∞ f ( x ) = ∞ , lim x → a + f ( x ) = − ∞ . ( \ displaystyle \ lim _ ( x \ to \ infty ) f ( x ) = \ infty , \ lim _ ( x \ to a ^ ( + ) ) f ( x ) = - \ infty . \ , ) For example lim x → 0 + ln  x = − ∞ . ( \ displaystyle \ lim _ ( x \ to 0 ^ ( + ) ) \ ln x = - \ infty . \ , ) Limits involving infinity are connected with the concept of asymptotes . These notions of a limit attempt to provide a metric space interpretation to limits at infinity . However , note that these notions of a limit are consistent with the topological space definition of limit if a neighborhood of − ∞ is defined to contain an interval ( − ∞ , c ) for some c ∈ R a neighborhood of ∞ is defined to contain an interval ( c , ∞ ) where c ∈ R a neighborhood of a ∈ R is defined in the normal way metric space R In this case , R is a topological space and any function of the form f : X → Y with X , Y ⊆ R is subject to the topological definition of a limit . Note that with this topological definition , it is easy to define infinite limits at finite points , which have not been defined above in the metric sense . Alternative notation ( edit ) Many authors allow for the projectively extended real line to be used as a way to include infinite values as well as extended real line . With this notation , the extended real line is given as R ∪ ( − ∞ , + ∞ ) and the projectively extended real line is R ∪ ( ∞ ) where a neighborhood of ∞ is a set of the form ( x : x > c ) . The advantage is that one only needs three definitions for limits ( left , right , and central ) to cover all the cases . As presented above , for a completely rigorous account , we would need to consider 15 separate cases for each combination of infinities ( five directions : − ∞ , left , central , right , and + ∞ ; three bounds : − ∞ , finite , or + ∞ ) . There are also noteworthy pitfalls . For example , when working with the extended real line , x − 1 ( \ displaystyle x ^ ( - 1 ) ) does not possess a central limit ( which is normal ) : lim x → 0 + 1 x = + ∞ , lim x → 0 − 1 x = − ∞ . ( \ displaystyle \ lim _ ( x \ to 0 ^ ( + ) ) ( 1 \ over x ) = + \ infty , \ lim _ ( x \ to 0 ^ ( - ) ) ( 1 \ over x ) = - \ infty . ) In contrast , when working with the projective real line , infinities ( much like 0 ) are unsigned , so , the central limit does exist in that context : lim x → 0 + 1 x = lim x → 0 − 1 x = lim x → 0 1 x = ∞ . ( \ displaystyle \ lim _ ( x \ to 0 ^ ( + ) ) ( 1 \ over x ) = \ lim _ ( x \ to 0 ^ ( - ) ) ( 1 \ over x ) = \ lim _ ( x \ to 0 ) ( 1 \ over x ) = \ infty . ) In fact there are a plethora of conflicting formal systems in use . In certain applications of numerical differentiation and integration , it is , for example , convenient to have signed zeroes . A simple reason has to do with the converse of lim x → 0 − x − 1 = − ∞ ( \ displaystyle \ lim _ ( x \ to 0 ^ ( - ) ) ( x ^ ( - 1 ) ) = - \ infty ) , namely , it is convenient for lim x → − ∞ x − 1 = − 0 ( \ displaystyle \ lim _ ( x \ to - \ infty ) ( x ^ ( - 1 ) ) = - 0 ) to be considered true . Such zeroes can be seen as an approximation to infinitesimals . Limits at infinity for rational functions ( edit ) Horizontal asymptote about y = 4 There are three basic rules for evaluating limits at infinity for a rational function f ( x ) = p ( x ) / q ( x ) : ( where p and q are polynomials ) : If the degree of p is greater than the degree of q , then the limit is positive or negative infinity depending on the signs of the leading coefficients ; If the degree of p and q are equal , the limit is the leading coefficient of p divided by the leading coefficient of q ; If the degree of p is less than the degree of q , the limit is 0 . If the limit at infinity exists , it represents a horizontal asymptote at y = L. Polynomials do not have horizontal asymptotes ; such asymptotes may however occur with rational functions . Functions of more than one variable ( edit ) By noting that x − p represents a distance , the definition of a limit can be extended to functions of more than one variable . In the case of a function f : R → R , lim ( x , y ) → ( p , q ) f ( x , y ) = L ( \ displaystyle \ lim _ ( ( x , y ) \ to ( p , q ) ) f ( x , y ) = L ) if for every ε > 0 there exists a δ > 0 such that for all ( x , y ) with 0 < ( x , y ) − ( p , q ) < δ , then f ( x , y ) − L < ε where ( x , y ) − ( p , q ) represents the Euclidean distance . This can be extended to any number of variables . Sequential limits ( edit ) Let f : X → Y be a mapping from a topological space X into a Hausdorff space Y , p ∈ X a limit point of X and L ∈ Y . The sequential limit of f as x tends to p is L if , for every sequence ( x ) in X − ( p ) that converges to p , the sequence f ( x ) converges to L . If L is the limit ( in the sense above ) of f as x approaches p , then it is a sequential limit as well , however the converse need not hold in general . If in addition X is metrizable , then L is the sequential limit of f as x approaches p if and only if it is the limit ( in the sense above ) of f as x approaches p . Other characterizations ( edit ) In terms of sequences ( edit ) For functions on the real line , one way to define the limit of a function is in terms of the limit of sequences . ( This definition is usually attributed to Eduard Heine . ) In this setting : lim x → a f ( x ) = L ( \ displaystyle \ lim _ ( x \ to a ) f ( x ) = L ) if and only if for all sequences x n ( \ displaystyle x_ ( n ) ) ( with x n ( \ displaystyle x_ ( n ) ) not equal to a for all n ) converging to a ( \ displaystyle a ) the sequence f ( x n ) ( \ displaystyle f ( x_ ( n ) ) ) converges to L ( \ displaystyle L ) . It was shown by Sierpiński in 1916 that proving the equivalence of this definition and the definition above , requires and is equivalent to a weak form of the axiom of choice . Note that defining what it means for a sequence x n ( \ displaystyle x_ ( n ) ) to converge to a ( \ displaystyle a ) requires the epsilon , delta method . Similarly as it was the case of Weierstrass 's definition , a more general Heine definition applies to functions defined on subsets of the real line . Let f be a real - valued function with the domain Dm ( f ) . Let a be the limit of a sequence of elements of Dm ( f ) \ ( a ) . Then the limit ( in this sense ) of f is L as x approaches p if for every sequence x n ( \ displaystyle x_ ( n ) ) ∈ Dm ( f ) \ ( a ) ( so that for all n , x n ( \ displaystyle x_ ( n ) ) is not equal to a ) that converges to a , the sequence f ( x n ) ( \ displaystyle f ( x_ ( n ) ) ) converges to L ( \ displaystyle L ) . This is the same as the definition of a sequential limit in the preceding section obtained by regarding the subset Dm ( f ) of R as a metric space with the induced metric . In non-standard calculus ( edit ) In non-standard calculus the limit of a function is defined by : lim x → a f ( x ) = L ( \ displaystyle \ lim _ ( x \ to a ) f ( x ) = L ) if and only if for all x ∈ R ∗ ( \ displaystyle x \ in \ mathbb ( R ) ^ ( * ) ) , f ∗ ( x ) − L ( \ displaystyle f ^ ( * ) ( x ) - L ) is infinitesimal whenever x − a ( \ displaystyle x-a ) is infinitesimal . Here R ∗ ( \ displaystyle \ mathbb ( R ) ^ ( * ) ) are the hyperreal numbers and f ∗ ( \ displaystyle f ^ ( * ) ) is the natural extension of f to the non-standard real numbers . Keisler proved that such a hyperreal definition of limit reduces the quantifier complexity by two quantifiers . On the other hand , Hrbacek writes that for the definitions to be valid for all hyperreal numbers they must implicitly be grounded in the ε - δ method , and claims that , from the pedagogical point of view , the hope that non-standard calculus could be done without ε - δ methods can not be realized in full . Bl·aszczyk et al. detail the usefulness of microcontinuity in developing a transparent definition of uniform continuity , and characterize Hrbacek 's criticism as a `` dubious lament '' . In terms of nearness ( edit ) At the 1908 international congress of mathematics F. Riesz introduced an alternate way defining limits and continuity in concept called `` nearness '' . A point x ( \ displaystyle x ) is defined to be near a set A ⊆ R ( \ displaystyle A \ subseteq \ mathbb ( R ) ) if for every r > 0 ( \ displaystyle r > 0 ) there is a point a ∈ A ( \ displaystyle a \ in A ) so that x − a < r ( \ displaystyle x-a < r ) . In this setting the lim x → a f ( x ) = L ( \ displaystyle \ lim _ ( x \ to a ) f ( x ) = L ) if and only if for all A ⊆ R ( \ displaystyle A \ subseteq \ mathbb ( R ) ) , L ( \ displaystyle L ) is near f ( A ) ( \ displaystyle f ( A ) ) whenever a ( \ displaystyle a ) is near A ( \ displaystyle A ) . Here f ( A ) ( \ displaystyle f ( A ) ) is the set ( f ( x ) x ∈ A ) ( \ displaystyle \ ( f ( x ) x \ in A \ ) ) . This definition can also be extended to metric and topological spaces . Relationship to continuity ( edit ) The notion of the limit of a function is very closely related to the concept of continuity . A function ƒ is said to be continuous at c if it is both defined at c and its value at c equals the limit of f as x approaches c : lim x → c f ( x ) = f ( c ) . ( \ displaystyle \ lim _ ( x \ to c ) f ( x ) = f ( c ) . ) ( We have here assumed that c is a limit point of the domain of f . ) Properties ( edit ) If a function f is real - valued , then the limit of f at p is L if and only if both the right - handed limit and left - handed limit of f at p exist and are equal to L . The function f is continuous at p if and only if the limit of f ( x ) as x approaches p exists and is equal to f ( p ) . If f : M → N is a function between metric spaces M and N , then it is equivalent that f transforms every sequence in M which converges towards p into a sequence in N which converges towards f ( p ) . If N is a normed vector space , then the limit operation is linear in the following sense : if the limit of f ( x ) as x approaches p is L and the limit of g ( x ) as x approaches p is P , then the limit of f ( x ) + g ( x ) as x approaches p is L + P. If a is a scalar from the base field , then the limit of af ( x ) as x approaches p is aL . If f is a real - valued ( or complex - valued ) function , then taking the limit is compatible with the algebraic operations , provided the limits on the right sides of the equations below exist ( the last identity only holds if the denominator is non-zero ) . This fact is often called the algebraic limit theorem . lim x → p ( f ( x ) + g ( x ) ) = lim x → p f ( x ) + lim x → p g ( x ) lim x → p ( f ( x ) − g ( x ) ) = lim x → p f ( x ) − lim x → p g ( x ) lim x → p ( f ( x ) ⋅ g ( x ) ) = lim x → p f ( x ) ⋅ lim x → p g ( x ) lim x → p ( f ( x ) / g ( x ) ) = lim x → p f ( x ) / lim x → p g ( x ) ( \ displaystyle ( \ begin ( matrix ) \ lim \ limits _ ( x \ to p ) & ( f ( x ) + g ( x ) ) & = & \ lim \ limits _ ( x \ to p ) f ( x ) + \ lim \ limits _ ( x \ to p ) g ( x ) \ \ \ lim \ limits _ ( x \ to p ) & ( f ( x ) - g ( x ) ) & = & \ lim \ limits _ ( x \ to p ) f ( x ) - \ lim \ limits _ ( x \ to p ) g ( x ) \ \ \ lim \ limits _ ( x \ to p ) & ( f ( x ) \ cdot g ( x ) ) & = & \ lim \ limits _ ( x \ to p ) f ( x ) \ cdot \ lim \ limits _ ( x \ to p ) g ( x ) \ \ \ lim \ limits _ ( x \ to p ) & ( f ( x ) / g ( x ) ) & = & ( \ lim \ limits _ ( x \ to p ) f ( x ) / \ lim \ limits _ ( x \ to p ) g ( x ) ) \ end ( matrix ) ) ) In each case above , when the limits on the right do not exist , or , in the last case , when the limits in both the numerator and the denominator are zero , nonetheless the limit on the left , called an indeterminate form , may still exist -- this depends on the functions f and g . These rules are also valid for one - sided limits , for the case p = ± ∞ , and also for infinite limits using the rules q + ∞ = ∞ for q ≠ − ∞ q × ∞ = ∞ if q > 0 q × ∞ = − ∞ if q < 0 q / ∞ = 0 if q ≠ ± ∞ ( see extended real number line ) . Note that there is no general rule for the case q / 0 ; it all depends on the way 0 is approached . Indeterminate forms -- for instance , 0 / 0 , 0 × ∞ , ∞ − ∞ , and ∞ / ∞ -- are also not covered by these rules , but the corresponding limits can often be determined with L'Hôpital's rule or the Squeeze theorem . Limits of compositions of functions ( edit ) In general , from knowing that lim y → b f ( y ) = c ( \ displaystyle \ lim _ ( y \ to b ) f ( y ) = c ) and lim x → a g ( x ) = b ( \ displaystyle \ lim _ ( x \ to a ) g ( x ) = b ) , it does not follow that lim x → a f ( g ( x ) ) = c ( \ displaystyle \ lim _ ( x \ to a ) f ( g ( x ) ) = c ) . However , this `` chain rule '' does hold if one of the following additional conditions holds : f ( b ) = c ( that is , f is continuous at b ) , or g does not take the value b near a ( that is , there exists a δ > 0 ( \ displaystyle \ delta > 0 ) such that if 0 < x − a < δ ( \ displaystyle 0 < x-a < \ delta ) then g ( x ) − b > 0 ( \ displaystyle g ( x ) - b > 0 ) ) . As an example of this phenomenon , consider the following functions that violates both additional restrictions : f ( x ) = g ( x ) = ( 0 if x ≠ 0 1 if x = 0 . ( \ displaystyle f ( x ) = g ( x ) = ( \ begin ( cases ) 0& ( \ text ( if ) ) x \ neq 0 \ \ 1& ( \ text ( if ) ) x = 0 \ end ( cases ) ) . ) Since the value at f ( 0 ) is a removable discontinuity , lim x → a f ( x ) = 0 ( \ displaystyle \ lim _ ( x \ to a ) f ( x ) = 0 ) for all a ( \ displaystyle a ) . Thus , the naïve chain rule would suggest that the limit of f ( f ( x ) ) is 0 . However , it is the case that f ( f ( x ) ) = ( 1 if x ≠ 0 0 if x = 0 ( \ displaystyle f ( f ( x ) ) = ( \ begin ( cases ) 1& ( \ text ( if ) ) x \ neq 0 \ \ 0& ( \ text ( if ) ) x = 0 \ end ( cases ) ) ) and so lim x → a f ( f ( x ) ) = 1 ( \ displaystyle \ lim _ ( x \ to a ) f ( f ( x ) ) = 1 ) for all a ( \ displaystyle a ) . Limits of special interest ( edit ) Rational functions ( edit ) For n ( \ displaystyle n ) a nonnegative integer and constants a 1 , a 2 , a 3 , ... , a n ( \ displaystyle a_ ( 1 ) , a_ ( 2 ) , a_ ( 3 ) , \ ldots , a_ ( n ) ) and b 1 , b 2 , b 3 , ... , b n ( \ displaystyle b_ ( 1 ) , b_ ( 2 ) , b_ ( 3 ) , \ ldots , b_ ( n ) ) , lim x → ∞ a 1 x n + a 2 x n − 1 + a 3 x n − 2 + ... + a n b 1 x n + b 2 x n − 1 + b 3 x n − 2 + ... + b n = a 1 b 1 ( \ displaystyle \ lim _ ( x \ to \ infty ) ( \ frac ( a_ ( 1 ) ( x ) ^ ( n ) + a_ ( 2 ) ( x ) ^ ( n - 1 ) + a_ ( 3 ) ( x ) ^ ( n - 2 ) + ... + a_ ( n ) ) ( b_ ( 1 ) ( x ) ^ ( n ) + b_ ( 2 ) ( x ) ^ ( n - 1 ) + b_ ( 3 ) ( x ) ^ ( n - 2 ) + ... + b_ ( n ) ) ) = ( \ frac ( a_ ( 1 ) ) ( b_ ( 1 ) ) ) ) This can be proven by dividing both the numerator and denominator by x n ( \ displaystyle x ^ ( n ) ) . If the numerator is a polynomial of higher degree , the limit does not exist . If the denominator is of higher degree , the limit is 0 . Trigonometric functions ( edit ) lim x → 0 sin  x x = 1 ( \ displaystyle \ lim _ ( x \ to 0 ) ( \ frac ( \ sin x ) ( x ) ) = 1 ) lim x → 0 1 − cos  x x = 0 ( \ displaystyle \ lim _ ( x \ to 0 ) ( \ frac ( 1 - \ cos x ) ( x ) ) = 0 ) lim x → ∞ x sin  ( 1 x ) = 1 ( \ displaystyle \ lim _ ( x \ to \ infty ) x \ sin \ left ( ( \ frac ( 1 ) ( x ) ) \ right ) = 1 ) Exponential functions ( edit ) lim x → 0 ( 1 + x ) 1 x = lim r → ∞ ( 1 + 1 r ) r = e ( \ displaystyle \ lim _ ( x \ to 0 ) ( 1 + x ) ^ ( \ frac ( 1 ) ( x ) ) = \ lim _ ( r \ to \ infty ) \ left ( 1 + ( \ frac ( 1 ) ( r ) ) \ right ) ^ ( r ) = e ) lim x → 0 e x − 1 x = 1 ( \ displaystyle \ lim _ ( x \ to 0 ) ( \ frac ( e ^ ( x ) - 1 ) ( x ) ) = 1 ) lim x → 0 e a x − 1 b x = a b ( \ displaystyle \ lim _ ( x \ to 0 ) ( \ frac ( e ^ ( ax ) - 1 ) ( bx ) ) = ( \ frac ( a ) ( b ) ) ) lim x → 0 c a x − 1 b x = a b ln  c ( \ displaystyle \ lim _ ( x \ to 0 ) ( \ frac ( c ^ ( ax ) - 1 ) ( bx ) ) = ( \ frac ( a ) ( b ) ) \ ln c ) lim x → 0 + x x = 1 ( \ displaystyle \ lim _ ( x \ to 0 ^ ( + ) ) x ^ ( x ) = 1 ) Logarithmic functions ( edit ) lim x → 0 ln  ( 1 + x ) x = 1 ( \ displaystyle \ lim _ ( x \ to 0 ) ( \ frac ( \ ln ( 1 + x ) ) ( x ) ) = 1 ) lim x → 0 ln  ( 1 + a x ) b x = a b ( \ displaystyle \ lim _ ( x \ to 0 ) ( \ frac ( \ ln ( 1 + ax ) ) ( bx ) ) = ( \ frac ( a ) ( b ) ) ) lim x → 0 log c  ( 1 + a x ) b x = a b ln  c ( \ displaystyle \ lim _ ( x \ to 0 ) ( \ frac ( \ log _ ( c ) ( 1 + ax ) ) ( bx ) ) = ( \ frac ( a ) ( b \ ln c ) ) ) L'Hôpital's rule ( edit ) Main article : l'Hôpital's rule This rule uses derivatives to find limits of indeterminate forms 0 / 0 or ± ∞ / ∞ , and only applies to such cases . Other indeterminate forms may be manipulated into this form . Given two functions f ( x ) and g ( x ) , defined over an open interval I containing the desired limit point c , then if : lim x → c f ( x ) = lim x → c g ( x ) = 0 , ( \ displaystyle \ lim _ ( x \ to c ) f ( x ) = \ lim _ ( x \ to c ) g ( x ) = 0 , ) or lim x → c f ( x ) = ± lim x → c g ( x ) = ± ∞ ( \ displaystyle \ lim _ ( x \ to c ) f ( x ) = \ pm \ lim _ ( x \ to c ) g ( x ) = \ pm \ infty ) , and f ( \ displaystyle f ) and g ( \ displaystyle g ) are differentiable over I ∖ ( c ) ( \ displaystyle I \ setminus \ ( c \ ) ) , and g ′ ( x ) ≠ 0 ( \ displaystyle g ' ( x ) \ neq 0 ) for all x ∈ I ∖ ( c ) ( \ displaystyle x \ in I \ setminus \ ( c \ ) ) , and lim x → c f ′ ( x ) g ′ ( x ) ( \ displaystyle \ lim _ ( x \ to c ) ( \ frac ( f ' ( x ) ) ( g ' ( x ) ) ) ) exists , then : lim x → c f ( x ) g ( x ) = lim x → c f ′ ( x ) g ′ ( x ) ( \ displaystyle \ lim _ ( x \ to c ) ( \ frac ( f ( x ) ) ( g ( x ) ) ) = \ lim _ ( x \ to c ) ( \ frac ( f ' ( x ) ) ( g ' ( x ) ) ) ) Normally , the first condition is the most important one . For example : lim x → 0 sin  ( 2 x ) sin  ( 3 x ) = lim x → 0 2 cos  ( 2 x ) 3 cos  ( 3 x ) = 2 ⋅ 1 3 ⋅ 1 = 2 3 . ( \ displaystyle \ lim _ ( x \ to 0 ) ( \ frac ( \ sin ( 2x ) ) ( \ sin ( 3x ) ) ) = \ lim _ ( x \ to 0 ) ( \ frac ( 2 \ cos ( 2x ) ) ( 3 \ cos ( 3x ) ) ) = ( \ frac ( 2 \ cdot 1 ) ( 3 \ cdot 1 ) ) = ( \ frac ( 2 ) ( 3 ) ) . ) Summations and integrals ( edit ) Specifying an infinite bound on a summation or integral is a common shorthand for specifying a limit . A short way to write the limit lim n → ∞ ∑ i = s n f ( i ) ( \ displaystyle \ lim _ ( n \ to \ infty ) \ sum _ ( i = s ) ^ ( n ) f ( i ) ) is ∑ i = s ∞ f ( i ) ( \ displaystyle \ sum _ ( i = s ) ^ ( \ infty ) f ( i ) ) . An important example of limits of sums such as these are series . A short way to write the limit lim x → ∞ ∫ a x f ( t ) d t ( \ displaystyle \ lim _ ( x \ to \ infty ) \ int _ ( a ) ^ ( x ) f ( t ) \ ; dt ) is ∫ a ∞ f ( t ) d t ( \ displaystyle \ int _ ( a ) ^ ( \ infty ) f ( t ) \ ; dt ) . A short way to write the limit lim x → − ∞ ∫ x b f ( t ) d t ( \ displaystyle \ lim _ ( x \ to - \ infty ) \ int _ ( x ) ^ ( b ) f ( t ) \ ; dt ) is ∫ − ∞ b f ( t ) d t ( \ displaystyle \ int _ ( - \ infty ) ^ ( b ) f ( t ) \ ; dt ) . See also ( edit ) List of limits One - sided limit Limit of a sequence Net ( topology ) Big O notation Limit superior and limit inferior l'Hôpital's rule Squeeze theorem Non-standard calculus Notes ( edit ) Jump up ^ For example , `` Limit '' at Encyclopaedia of Mathematics Jump up ^ Keisler , H. Jerome ( 2008 ) , `` Quantifiers in limits '' ( PDF ) , Andrzej Mostowski and foundational studies , IOS , Amsterdam , pp. 151 -- 170 Jump up ^ Hrbacek , K. ( 2007 ) , `` Stratified Analysis ? '' , in Van Den Berg , I. ; Neves , V. , The Strength of Nonstandard Analysis , Springer Jump up ^ Bl·aszczyk , Piotr ; Katz , Mikhail ; Sherry , David ( 2012 ) , `` Ten misconceptions from the history of analysis and their debunking '' , Foundations of Science , 18 ( 1 ) , arXiv : 1202.4153 , doi : 10.1007 / s10699 - 012 - 9285 - 8 References ( edit ) MacTutor History of Weierstrass . MacTutor History of Bolzano Visual Calculus by Lawrence S. Husch , University of Tennessee ( 2001 ) Apostol , Tom M. , Mathematical Analysis , 2nd ed . Addison -- Wesley , 1974 . ISBN 0 - 201 - 00288 - 4 . Bartle , Robert ( 1967 ) , The elements of real analysis , Wiley Burton , David M. ( 1997 ) , The History of Mathematics : An introduction ( Third ed . ) , New York : McGraw -- Hill , pp. 558 -- 559 , ISBN 0 - 07 - 009465 - 9 Felscher , Walter ( 2000 ) , `` Bolzano , Cauchy , Epsilon , Delta '' , American Mathematical Monthly , Mathematical Association of America , 107 ( 9 ) : 844 -- 862 , doi : 10.2307 / 2695743 , JSTOR 2695743 . Courant , Richard ( 1924 ) , Vorlesungen über Differential - und Integralrechnung , Springer Verlag Grabiner , Judith V. ( 1983 ) , `` Who Gave You the Epsilon ? Cauchy and the Origins of Rigorous Calculus '' , American Mathematical Monthly , Mathematical Association of America , 90 ( 3 ) : 185 -- 194 , doi : 10.2307 / 2975545 , JSTOR 2975545 , collected in Who Gave You the Epsilon ? ISBN 978 - 0 - 88385 - 569 - 0 pp. 5 -- 13 . Also available at : http://www.maa.org/pubs/Calc_articles/ma002.pdf Hardy , G.H. ( 1921 ) , A course in pure mathematics , Cambridge University Press Hubbard , John H. ( 2015 ) , Vector calculus , linear algebra , and differential forms : A unified approach ( Fifth ed . ) , Matrix Editions Miller , Jeff ( 1 December 2004 ) , Earliest Uses of Symbols of Calculus , retrieved 2008 - 12 - 18 . Page , Warren ; Hersh , Reuben ; Selden , Annie ; et al. , eds. ( 2002 ) , `` Media Highlights '' , The College Mathematics , Mathematical Association of America , 33 ( 2 ) : 147 -- 154 , JSTOR Journal 2687124 Journal . Sutherland , W.A. , Introduction to Metric and Topological Spaces . Oxford University Press , Oxford , 1975 . ISBN 0 - 19 - 853161 - 3 . Rudin , Walter ( 1964 ) , Principles of mathematical analysis , McGraw - Hill Whittaker ; Watson ( 1904 ) , A course of modern analysis , Cambridge University Press Retrieved from `` https://en.wikipedia.org/w/index.php?title=Limit_of_a_function&oldid=837083727 '' Categories : Limits ( mathematics ) Functions and mappings Hidden categories : Use dmy dates from September 2010 Talk Contents About Wikipedia Afrikaans Беларуская Беларуская ( тарашкевіца )  Bosanski Čeština Deutsch Eesti Ελληνικά Español فارسی Français Galego 한국어 हिन्दी Bahasa Indonesia Italiano עברית ქართული Македонски Bahasa Melayu 日本 語 Polski Português Română Русский සිංහල Simple English Slovenščina Српски / srpski Suomi Tagalog தமிழ் ไทย Türkçe Українська اردو 粵語 中文 30 more Edit links This page was last edited on 18 April 2018 , at 16 : 22 . Text is available under the Creative Commons Attribution - ShareAlike License ; additional terms may apply . By using this site , you agree to the Terms of Use and Privacy Policy . 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Question: when does a limit of a function not exist


I Do n't Like Mondays - wikipedia I Do n't Like Mondays Jump to : navigation , search `` I Do n't Like Mondays '' Single by The Boomtown Rats from the album The Fine Art of Surfacing B - side `` It 's All the Rage '' Released 21 July 1979 ( UK ) Format 7 '' vinyl Recorded Trident Studios Length 4 : 19 ( LP ) 3 : 47 ( single / video ) Label Ensign ( UK ) Columbia ( US ) Songwriter ( s ) Bob Geldof Producer ( s ) Phil Wainman The Boomtown Rats singles chronology `` Rat Trap '' ( 1978 ) `` I Do n't Like Mondays '' ( 1979 ) `` Diamond Smiles '' ( 1979 ) `` Rat Trap '' ( 1978 ) `` I Do n't Like Mondays '' ( 1979 ) `` Diamond Smiles '' ( 1979 ) `` I Do n't Like Mondays '' is a song by Irish band The Boomtown Rats that was a number one single in the UK Singles Chart for four weeks during the summer of 1979 , and ranks as the sixth biggest British hit of 1979 . Written by Bob Geldof , it was the band 's second number one single . The full length version appeared on the group 's third album , The Fine Art of Surfacing . It includes a reprise of the first verse , which was edited for the single release . Contents ( hide ) 1 History 2 Chart performance 2.1 Weekly singles charts 2.2 Year - end charts 3 Live performances 4 References History ( edit ) According to Geldof , he wrote the song after reading a telex report at Georgia State University 's campus radio station , WRAS , on the shooting spree of 16 - year - old Brenda Ann Spencer , who fired at children in a school playground at Grover Cleveland Elementary School in San Diego , California , US on 29 January 1979 , killing two adults and injuring eight children and one police officer . Spencer showed no remorse for her crime and her full explanation for her actions was `` I do n't like Mondays . This livens up the day '' . The song was first performed less than a month later . Geldof explained how he wrote the song : I was doing a radio interview in Atlanta with ( Johnnie ) Fingers and there was a telex machine beside me . I read it as it came out . Not liking Mondays as a reason for doing somebody in is a bit strange . I was thinking about it on the way back to the hotel and I just said ' Silicon chip inside her head had switched to overload ' . I wrote that down . And the journalists interviewing her said , ' Tell me why ? ' It was such a senseless act . It was the perfect senseless act and this was the perfect senseless reason for doing it . So perhaps I wrote the perfect senseless song to illustrate it . It was n't an attempt to exploit tragedy . Geldof had originally intended the song as a B - side , but changed his mind after the song was successful with audiences on the Rats ' US tour . Spencer 's family tried unsuccessfully to prevent the single from being released in the United States . Despite being a major hit in the United Kingdom , it only reached # 73 on the US Billboard Hot 100 . The song was played regularly by album - oriented rock format radio stations in the United States throughout the 1980s , although radio stations in San Diego refrained from playing the track for some years in respect to local sensitivities about the shooting . The strings were arranged by Fiachra Trench . Apart from Geldof , pianist Johnnie Fingers and drummer Simon Crowe are the only members of the group who appear on the recording . In the UK the song reached Number One on the singles chart in July 1979 , and won the Best Pop Song and Outstanding British Lyric categories at the Ivor Novello Awards . Chart performance ( edit ) Weekly singles charts ( edit ) Chart ( 1979 ) Peak position Australia Canada Ireland New Zealand UK U.S. Billboard Hot 100 73 U.S. Cash Box Top 100 84 Year - end charts ( edit ) Chart ( 1979 ) Position Australia 6 Canada 117 New Zealand 25 UK Chart ( 1980 ) Position Canada 55 Live performances ( edit ) On 9 September 1981 , Geldof was joined on stage by fellow Boomtown Rat , Johnnie Fingers , to perform the song for The Secret Policeman 's Ball sponsored by Amnesty International . A recording of that performance appears on the 1982 album , The Secret Policeman 's Other Ball . The Boomtown Rats performed the song for Live Aid at Wembley Stadium in 1985 . This was the band 's final major appearance . On singing the line , `` And the lesson today is how to die '' , Geldof paused for 20 seconds while the crowd applauded on the significance to those starving in Africa that Live Aid was intended to help . At a concert in London in 1995 , almost ten years later to the day , Bon Jovi covered the song after being joined on stage by Geldof at Wembley Stadium . This recorded performance features on Bon Jovi 's live album One Wild Night Live 1985 -- 2001 , as well as on the bonus 2 - CD edition of These Days . Bon Jovi was again joined by Geldof for a performance of the song at The O2 Arena on 23 June 2010 , the 10th night of their 12 - night residency . Geldof himself performed a version of the song while hosting the Live 8 concert in London , on 2 July 2005 . `` I Do n't Like Mondays '' was subsequently covered by Tori Amos on her 2001 album Strange Little Girls and later by G4 on their 2006 album Act Three . References ( edit ) Jump up ^ Roberts , David ( 2006 ) . British Hit Singles & Albums ( 19th ed . ) . London : Guinness World Records Limited . pp. 370 -- 1 . ISBN 1 - 904994 - 10 - 5 . Jump up ^ `` Top 100 1979 - UK Music Charts '' . Uk-charts.top-source.info . Retrieved 2015 - 10 - 20 . Jump up ^ Mikkelson , Barbara ( 29 September 2005 ) . `` Urban Legends Reference Pages : Music ( I Do n't Like Mondays ) '' . snopes.com . ^ Jump up to : Clarke , Steve ( 18 -- 31 October 1979 ) . The Fastest Lip on Vinyl . Smash Hits . EMAP National Publications Ltd . pp. 6 -- 7 . Jump up ^ Whitburn , Joel ( 2000 ) . Top Pop Singles 1955 - 1999 . Record Research Inc. p. 65 . ISBN 0 - 89820 - 139 - X . Jump up ^ `` I Do n't Like Mondays '' . Songfacts.com . Retrieved 2009 - 04 - 05 . ^ Jump up to : Steffen Hung . `` Forum - 1970 ( ARIA Charts : Special Occasion Charts ) '' . australian-charts.com . Archived from the original on 2 June 2016 . Retrieved 20 October 2015 . Jump up ^ `` Image : RPM Weekly - Library and Archives Canada '' . Bac-lac.gc.ca . Retrieved 2015 - 10 - 20 . Jump up ^ `` Image : RPM Weekly - Library and Archives Canada '' . Bac-lac.gc.ca . Retrieved 2015 - 10 - 20 . Jump up ^ `` Top Selling Singles of 1979 The Official New Zealand Music Chart '' . Nztop40.co.nz. 1979 - 12 - 31 . Retrieved 2016 - 10 - 15 . Jump up ^ `` Top 100 Singles of 1979 '' Record Mirror 5 January 1980 : 30 Preceded by `` Are ' Friends ' Electric ? '' by Tubeway Army UK number - one single 28 July 1979 - 24 August 1979 Succeeded by `` We Do n't Talk Anymore '' by Cliff Richard Preceded by `` Do You Want Your Oul Lobby Washed Down '' by Brendan Shine Irish Singles Chart number - one single 4 August 1979 - 31 August 1979 Succeeded by `` Viva Il Papa '' by Catriona Walsh Preceded by `` Born to Be Alive '' by Patrick Hernandez Australian Kent Music Report number - one single 12 November 1979 - 25 November 1979 Succeeded by `` Video Killed the Radio Star '' by The Buggles ( hide ) The Boomtown Rats Bob Geldof Pete Briquette Gerry Cott Simon Crowe Johnnie Fingers Garry Roberts Studio albums The Boomtown Rats A Tonic for the Troops The Fine Art of Surfacing Mondo Bongo V Deep In the Long Grass Singles `` Lookin ' After No. 1 '' `` Mary of the 4th Form '' `` She 's So Modern '' `` Like Clockwork '' `` Rat Trap '' `` I Do n't Like Mondays '' `` Diamond Smiles '' `` Someone 's Looking at You '' `` Banana Republic '' `` Up All Night '' `` The Elephant 's Graveyard ( Guilty ) '' `` Never in a Million Years '' `` House on Fire '' `` Charmed Lives '' `` Tonight '' `` Drag Me Down '' `` Dave '' `` A Hold of Me '' Other albums Ratrospective The Boomtown Rats ' Greatest Hits Loudmouth -- The Best of Bob Geldof and the Boomtown Rats The Best of The Boomtown Rats Back to Boomtown : Classic Rats Hits So Modern -- The Boomtown Rats Collection Related articles Discography Ensign Records Mercury Records Retrieved from `` https://en.wikipedia.org/w/index.php?title=I_Don%27t_Like_Mondays&oldid=813165947 '' Categories : 1979 singles European Hot 100 Singles number - one singles Irish Singles Chart number - one singles Songs based on actual events Songs inspired by deaths Songs about school Songs about the media Songs written by Bob Geldof The Boomtown Rats songs UK Singles Chart number - one singles Song recordings produced by Phil Wainman 1979 songs Ensign Records singles Columbia Records singles Hidden categories : Articles with hAudio microformats All articles with unsourced statements Articles with unsourced statements from July 2011 Use dmy dates from July 2011 Talk Contents About Wikipedia Català Dansk Deutsch Español Français Italiano עברית Nederlands 日本 語 Norsk Português Русский Srpskohrvatski / српскохрватски Svenska Edit links This page was last edited on 2 December 2017 , at 04 : 55 . 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Question: when was i don't like mondays released


World economy - wikipedia World economy Jump to : navigation , search World economy Africa Americas Central America North America South America Asia Europe Oceania 20 Largest economies pie chart The world economy or global economy is the economy of the world , considered as the international exchange of goods and services that is expressed in monetary units of account ( money ) . In some contexts , the two terms are distinguished : the `` international '' or `` global economy '' being measured separately and distinguished from national economies while the `` world economy '' is simply an aggregate of the separate countries ' measurements . Beyond the minimum standard concerning value in production , use , and exchange the definitions , representations , models , and valuations of the world economy vary widely . It is inseparable from the geography and ecology of Earth . It is common to limit questions of the world economy exclusively to human economic activity , and the world economy is typically judged in monetary terms , even in cases in which there is no efficient market to help valuate certain goods or services , or in cases in which a lack of independent research or government cooperation makes establishing figures difficult . Typical examples are illegal drugs and other black market goods , which by any standard are a part of the world economy , but for which there is by definition no legal market of any kind . However , even in cases in which there is a clear and efficient market to establish a monetary value , economists do not typically use the current or official exchange rate to translate the monetary units of this market into a single unit for the world economy , since exchange rates typically do not closely reflect worldwide value , for example in cases where the volume or price of transactions is closely regulated by the government . World share of GDP ( PPP ) ( World Bank , 2011 ) . Rather , market valuations in a local currency are typically translated to a single monetary unit using the idea of purchasing power . This is the method used below , which is used for estimating worldwide economic activity in terms of real US dollars or euros . However , the world economy can be evaluated and expressed in many more ways . It is unclear , for example , how many of the world 's 7.13 billion people have most of their economic activity reflected in these valuations . As of 2017 , the following 15 countries or regions have reached an economy of at least US $2 trillion by GDP in nominal or PPP terms : Brazil , China , France , Germany , India , Indonesia , Italy , Japan , South Korea , Mexico , Russia , Turkey , the United Kingdom , the United States , and the European Union . Contents ( hide ) 1 Economy -- overview 1.1 World economy by Country Groups 1.2 Current world economic league table of largest economies in the world by GDP and share of global economic growth 1.3 Twenty largest economies in the world by nominal GDP 1.4 Twenty Largest Economies in the World by PPP GDP ( IMF and CIA World Factbook ) 2 Statistical indicators 2.1 Economy 2.2 Employment 2.3 Industries 2.4 Energy 2.5 Cross-border 2.6 Gift economy 2.7 Communications 2.8 Transport 2.9 Military 3 Economic studies 4 See also 5 References 6 External links Economy -- overview ( edit ) World economy by country Groups ( edit ) The following two tables list the Country Groups with individual countries designated by the IMF . Members of the G - 20 major economies are in bold . List of Country Groups by GDP ( nominal ) in Millions US $ List of Country Groups by GDP ( PPP ) in 2015 in Millions US $ Country Group GDP ( Nominal ) Peak Year Number of Countries Economies with at least 0.5 % of Global GDP Major advanced economies ( G7 ) 36,006,539 2017 7 Canada France Germany Italy Japan United Kingdom United States Emerging and Developing Asia 17,084,823 2017 30 China India Indonesia Thailand Other Advanced Economies ( Advanced economies excluding G7 ) 11,593,086 2014 32 Australia Austria Belgium South Korea Netherlands Norway Spain Sweden Switzerland Taiwan Latin America and the Caribbean 5,983,936 2013 32 Argentina Brazil Mexico Middle East , North Africa , Afghanistan , and Pakistan 3,473,402 2014 22 Iran Saudi Arabia United Arab Emirates Commonwealth of Independent States and Georgia 2,943,338 2013 12 Russia Emerging and developing Europe 2,031,117 2014 12 Poland Turkey Sub-Saharan Africa 1,690,338 2014 45 Nigeria World 78,519,556 2014 192 Country Group GDP ( Nominal ) Peak Year Number of Countries Economies with at least 0.5 % of Global GDP Emerging and Developing Asia 41,233,893 2017 30 China India Indonesia Malaysia Philippines Thailand Major advanced economies ( G7 ) 38,766,514 2017 7 Canada France Germany Italy Japan United Kingdom United States Other Advanced Economies ( Advanced economies excluding G7 ) 13,587,681 2017 32 Australia South Korea Netherlands Spain Taiwan Latin America and the Caribbean 9,730,047 2017 32 Argentina Brazil Colombia Mexico Venezuela Middle East , North Africa , Afghanistan , and Pakistan 9,529,045 2017 22 Algeria Egypt Iran Pakistan Saudi Arabia United Arab Emirates Commonwealth of Independent States and Georgia 5,583,406 2017 12 Russia Emerging and developing Europe 4,453,006 2017 12 Poland Turkey Sub-Saharan Africa 3,804,491 2017 45 Nigeria South Africa World 126,688,083 2017 192 Current world economic league table of largest economies in the world by GDP and share of global economic growth ( edit ) The following two tables list the 25 largest economies by GDP ( Nominal ) , twenty largest economies by GDP ( PPP ) . Members of the G - 20 major economies are in bold . List of the 25 largest economies by GDP ( nominal ) at their peak level of GDP in Millions US $ List of the 25 largest economies by GDP ( PPP ) at their peak level of GDP in Millions US $ Rank Country Value ( US D $ ) Peak Year -- World 78,519,556 2014 United States 19,417,144 2017 -- European Union 19,200,287 2008 China 11,795,297 2017 Japan 6,203,213 2012 Germany 3,885,440 2014 5 United Kingdom 3,064,351 2007 6 France 2,937,321 2008 7 Brazil 2,614,027 2011 8 India 2,454,458 2017 9 Italy 2,402,062 2008 10 Russia 2,230,624 2013 11 Canada 1,842,627 2013 12 Spain 1,642,738 2008 13 Australia 1,509,639 2012 14 South Korea 1,498,074 2017 15 Mexico 1,298,404 2014 16 Indonesia 1,020,515 2017 17 Turkey 950,328 2013 18 Netherlands 940,665 2008 19 Saudi Arabia 756,350 2014 20 Switzerland 702,736 2014 21 Argentina 631,621 2015 22 Sweden 578,742 2013 23 Iran 570,039 2011 24 Nigeria 568,496 2014 25 Taiwan 566,757 2017 Rank Country Value ( US D $ ) Peak Year -- World 126,688,083 2017 China 23,194,411 2017 -- European Union 20,852,702 2017 United States 19,417,144 2017 India 9,489,302 2017 Japan 5,420,228 2017 5 Germany 4,134,668 2017 6 Russia 3,938,001 2017 7 Brazil 3,306,570 2014 8 Indonesia 3,257,123 2017 9 United Kingdom 2,905,392 2017 10 France 2,833,064 2017 11 Mexico 2,406,199 2017 12 Italy 2,303,108 2017 13 Turkey 2,082,079 2017 14 South Korea 2,029,706 2017 15 Saudi Arabia 1,796,205 2017 16 Spain 1,768,816 2017 17 Canada 1,752,910 2017 18 Iran 1,535,491 2017 19 Australia 1,251,416 2017 20 Thailand 1,226,407 2017 21 Egypt 1,197,967 2017 22 Taiwan 1,177,052 2017 23 Nigeria 1,124,627 2017 24 Poland 1,114,105 2017 25 Pakistan 1,060,569 2017 Twenty largest economies in the world by nominal GDP ( edit ) Main article : List of IMF ranked countries by historical GDP ( nominal ) The following is a list of the twenty largest economies by nominal GDP at peak value as of the specific year according to International Monetary Fund . Rank 1980 1985 1990 1995 2000 2005 2015 2020 United States United States United States United States United States United States United States United States United States Soviet Union Soviet Union Japan Japan Japan Japan China China China Japan Japan Soviet Union Germany Germany Germany Japan Japan Japan West Germany West Germany West Germany France United Kingdom United Kingdom Germany Germany Germany 5 France France France United Kingdom France China United Kingdom United Kingdom India 6 United Kingdom United Kingdom United Kingdom Italy Italy France France France United Kingdom 7 Italy Italy Italy Brazil China Italy Italy Brazil France 8 China Canada Canada China Brazil Canada Brazil Italy Brazil 9 Canada China Iran Spain Canada Spain Russia Russia Italy 10 Mexico India Spain Canada Mexico South Korea India India Russia Rank 1980 1985 1990 1995 2000 2005 2015 2020 11 Spain Brazil Brazil Iran Spain Brazil Spain Canada Canada 12 Brazil Mexico China South Korea South Korea Mexico Canada Spain South Korea 13 India Australia India Mexico Iran India Australia Australia Spain 14 Netherlands Spain Australia Netherlands India Russia South Korea South Korea Australia 15 Australia Iran Netherlands Australia Netherlands Australia Mexico Mexico Indonesia 16 Saudi Arabia Netherlands Mexico India Russia Netherlands Netherlands Turkey Mexico 17 Sweden Sweden South Korea Switzerland Australia Iran Turkey Netherlands Turkey 18 Belgium Saudi Arabia Switzerland Russia Switzerland Turkey Indonesia Indonesia Netherlands 19 Switzerland Switzerland Sweden Belgium Taiwan Switzerland Switzerland Saudi Arabia Saudi Arabia 20 Iran South Korea Turkey Argentina Argentina Sweden Iran Switzerland Argentina Twenty largest economies in the world by PPP GDP ( IMF and CIA world Factbook ) ( edit ) The following is a list of twenty largest economies by GDP ( PPP ) at peak value as of the specific year according to CIA World Factbook and the International Monetary Fund . Rank 1980 1985 1990 1995 2000 2005 2015 2020 United States United States United States United States United States United States United States China China Soviet Union Soviet Union Japan Japan China China China United States United States Japan Japan West Germany China Japan Japan India India India Italy West Germany Soviet Union Germany Germany India Japan Japan Japan 5 West Germany Italy Italy Russia India Germany Russia Germany Germany 6 France France France India Russia Russia Germany Russia Russia 7 Brazil Brazil China Italy France Brazil Brazil Brazil Indonesia 8 United Kingdom United Kingdom United Kingdom France Italy France France Indonesia Brazil 9 Mexico China Brazil Brazil Brazil United Kingdom United Kingdom United Kingdom United Kingdom 10 India India India United Kingdom United Kingdom Italy Italy France France Rank 1980 1985 1990 1995 2000 2005 2015 2020 11 China Mexico Mexico Mexico Mexico Mexico Indonesia Mexico Mexico 12 Spain Canada Canada Indonesia Indonesia Indonesia Mexico Italy Italy 13 Canada Spain Indonesia Spain Spain Spain Spain Turkey Turkey 14 Saudi Arabia Iran Spain Canada Canada Canada South Korea South Korea South Korea 15 Indonesia Indonesia Saudi Arabia Saudi Arabia South Korea South Korea Canada Saudi Arabia Saudi Arabia 16 Argentina Saudi Arabia Turkey South Korea Saudi Arabia Saudi Arabia Iran Canada Spain 17 Iran Turkey Iran Turkey Turkey Turkey Turkey Spain Canada 18 Netherlands Australia South Korea Iran Iran Iran Saudi Arabia Iran Iran 19 Turkey Netherlands Australia Australia Australia Australia Australia Australia Egypt 20 Poland Argentina Netherlands Thailand Netherlands Thailand Taiwan Thailand Australia Statistical indicators ( edit ) Economy ( edit ) GDP ( GWP ) ( gross world product ) : ( purchasing power parity exchange rates ) -- $59.38 trillion ( 2005 est . ) , $51.48 trillion ( 2004 ) , $23 trillion ( 2002 ) GDP ( GWP ) ( gross world product ) : ( market exchange rates ) -- $60.69 trillion ( 2008 ) GDP -- real growth rate : 3.2 % ( 2008 ) , 3.1 % p.a. ( 2000 -- 07 ) , 2.4 % p.a. ( 1990 -- 99 ) , 3.1 % p.a. ( 1980 -- 89 ) GDP -- per capita : purchasing power parity -- $9,300 , € 7,500 ( 2005 est . ) , $8,200 , € 6,800 ( 92 ) ( 2003 ) , $7,900 , € 5,000 ( 2002 ) World median income : purchasing power parity $1,041 , € 950 ( 1993 ) GDP -- composition by sector : agriculture : 4 % ; industry : 32 % ; services : 64 % ( 2004 est . ) Inflation rate ( consumer prices ) : developed countries 1 % to 4 % typically ; developing countries 5 % to 60 % typically ; national inflation rates vary widely in individual cases , from declining prices in Japan to hyperinflation in several Third World countries ( 2003 ) Derivatives OTC outstanding notional amount : $601 trillion ( Dec 2010 ) ( ( 4 ) ) Derivatives exchange traded outstanding notional amount : $82 trillion ( June 2011 ) ( ( 5 ) ) Global debt issuance : $5.187 trillion , € 3 trillion ( 2004 ) , $4.938 trillion , € 3.98 trillion ( 2003 ) , $3.938 trillion ( 2002 ) ( Thomson Financial League Tables ) Global equity issuance : $505 billion , € 450 billion ( 2004 ) , $388 billion . € 320 billion ( 2003 ) , $319 billion , € 250 trillion ( 2002 ) ( Thomson Financial League Tables ) Employment ( edit ) World GDP per capita between 1500 -- 2000 ( log scale ) World GDP per capita between 1500 -- 2003 GDP increase , 1990 -- 1998 and 1990 -- 2006 , in major countries . Unemployment rate : 8.7 % ( 2009 est . ) . 30 % ( 2007 est . ) combined unemployment and underemployment in many non-industrialized countries ; developed countries typically 4 % -- 12 % unemployment . Industries ( edit ) Industrial production growth rate : 3 % ( 2002 est . ) Energy ( edit ) Yearly electricity -- production : 21,080,878 GWh ( 2011 est . ) , 15,850,000 GWh ( 2003 est . ) , 14,850,000 GWh ( 2001 est . ) Yearly electricity -- consumption : 14,280,000 GWh ( 2003 est . ) , 13,930,000 GWh ( 2001 est . ) Oil -- production : 79,650,000 bbl / d ( 12,663,000 m / d ) ( 2003 est . ) , 75,460,000 barrels per day ( 11,997,000 m / d ) ( 2001 ) Oil -- consumption : 80,100,000 bbl / d ( 12,730,000 m / d ) ( 2003 est . ) , 76,210,000 barrels per day ( 12,116,000 m / d ) ( 2001 ) Oil -- proved reserves : 1.025 trillion barrel ( 163 km3 ) ( 2001 est . ) Natural gas -- production : 3,366 km3 ( 2012 est . ) , 2,569 km3 ( 2001 est . ) Natural gas -- consumption : 2,556 km3 ( 2001 est . ) Natural gas -- proved reserves : 161,200 km3 ( 1 January 2002 ) Cross-border ( edit ) Yearly exports : $12.4 trillion , € 11.05 trillion ( 2009 est . ) Exports -- commodities : the whole range of industrial and agricultural goods and services Exports -- partners : US 12.7 % , Germany 7.1 % , China 6.2 % , France 4.4 % , Japan 4.2 % , UK 4.1 % ( 2008 ) Yearly imports : $12.29 trillion , € 10.95 trillion ( 2009 est . ) Imports -- commodities : the whole range of industrial and agricultural goods and services Imports -- partners : China 10.3 % , Germany 8.6 % , US 8.1 % , Japan 5 % ( 2008 ) Debt -- external : $56.9 trillion , € 40 trillion ( 31 December 2009 est . ) Gift economy ( edit ) Yearly economic aid -- recipient : net Official Development Assistance ( ODA ) of $135.2 billion ( 2014 ) Communications ( edit ) Telephones -- main lines in use : 843,923,500 ( 2007 ) 4,263,367,600 ( 2008 ) Telephones -- mobile cellular : 3,300,000,000 ( Nov. 2007 ) Internet Service Providers ( ISPs ) : 10,350 ( 2000 est . ) Internet users : 3,079,339,857 ( December 31 , 2014 ( 6 ) ) , 360,985,492 ( December 31 , 2000 ) Transport ( edit ) Transportation infrastructure worldwide includes : Airports Total : 41,821 ( 2013 ) Roadways ( in kilometres ) Total : 32,345,165 km Paved : 19,403,061 km Unpaved : 12,942,104 km ( 2002 ) Railways Total : 1,122,650 km includes about 190,000 to 195,000 km of electrified routes of which 147,760 km are in Europe , 24,509 km in the Far East , 11,050 km in Africa , 4,223 km in South America , and 4,160 km in North America . Military ( edit ) World military expenditure in 2012 : estimated to $1.756 trillion Military expenditures -- percent of GDP : roughly 2 % of gross world product ( 1999 ) . Economic studies ( edit ) To promote exports , many government agencies publish on the web economic studies by sector and country . Among these agencies include the USCS ( US DoC ) and FAS ( USDA ) in the United States , EDC and AAFC in Canada , Ubifrance in France , UKTI in the UK , HKTDC and JETRO in Asia , Austrade and NZTE in Oceania . Through Partnership Agreements , The Federation of International Trade Associations publishes studies from several of these agencies ( USCS , FAS , AAFC , UKTI , HKTDC ) , as well as other non-governmental organizations on its website GlobalTrade.net . See also ( edit ) Capitalism Common Wealth : Economics for a Crowded Planet ( book ) Economic collapse Economic history of the world Emerging and growth - leading economies Global financial system Global workforce Globality Globalization Trade route Overconsumption World Trade Report World systems theory Regional economies : Economy of Africa Economy of Asia Economy of Europe Economy of North America Economy of Oceania Economy of South America Events : 2007 -- 2008 world food price crisis Late 2000s recession Oil price increases since 2003 Lists : List of countries by GDP sector composition List of world 's largest economies ( nominal ) -- based on current currency market exchange rates List of world 's largest economies ( PPP ) -- based on purchasing power parity Historical list of world 's largest economies ( PPP ) -- for the years between 1 and 1998 List of world production References ( edit ) Jump up ^ `` World Economy . '' -- Definition . American English Definition of with Pronunciation by Macmillan Dictionary . N.p. , n.d. Web . 02 Jan. 2015 . Jump up ^ `` The International Comparison Program ( ICP ) '' . Retrieved 1 April 2017 . Jump up ^ `` Gross domestic product , current prices & Gross domestic product based on purchasing - power - parity ( PPP ) valuation of country GDP '' . World Economic Outlook Database , October 2015 . International Monetary Fund . October 2015 . Retrieved 2015 - 10 - 15 . Jump up ^ ( 1 ) IMF GDP 2017 Data ( April 2017 ) ^ Jump up to : `` Gross domestic product '' . IMF World Economic Outlook . April 2017 . Retrieved 14 April 2017 . Jump up ^ ( 2 ) IMF GDP ( Nominal ) Data ( April 2017 ) Jump up ^ ( 3 ) IMF GDP ( PPP ) Data ( April 2017 ) Jump up ^ `` Gross domestic product , current prices '' . International Monetary Fund . International Monetary Fund . October 2015 . Retrieved 11 October 2015 . Jump up ^ `` Gross domestic product based on purchasing - power - parity ( PPP ) valuation of country GDP ; Current international dollar '' . International Monetary Fund . April 2017 . Retrieved 14 May 2017 . Jump up ^ `` GDP ( PURCHASING POWER PARITY ) '' . CIA World Factbook . CIA World Factbook. 2014 . Retrieved October 7 , 2014 . Jump up ^ `` IMF World Economic Outlook , Crisis and Recovery , April 2009 '' . Imf.org. 2009 - 04 - 16 . Retrieved 2012 - 07 - 17 . Jump up ^ B. Milanovic ( January 2002 ) . `` True World Income Distribution , 1988 and 1993 : First Calculation Based on Household Surveys Alone '' ( PDF ) . Retrieved 13 May 2011 . Jump up ^ `` International Energy Statistics '' . Retrieved 1 April 2017 . Jump up ^ `` International Energy Statistics '' . Retrieved 1 April 2017 . Jump up ^ `` Development aid stable in 2014 but flows to poorest countries still falling '' . OECD. 2015 - 04 - 08 . Retrieved 2015 - 07 - 23 . Jump up ^ global cellphone penetration reaches 50 percent Jump up ^ `` World Internet Usage Statistics News and World Population Stats '' . Internetworldstats.com . Retrieved 2015 - 06 - 08 . Jump up ^ `` The World Factbook -- Central Intelligence Agency '' . Retrieved 1 April 2017 . Jump up ^ `` 3 . Military expenditure - SIPRI '' . Retrieved 1 April 2017 . External links ( edit ) Wikiquote has quotations related to : World economy OECD -- Economic Outlook US Bureau of Labor and Statistics , Major Economic Indicators IMF -- World Economic Outlook UN DESA -- World Economy publications CIA -- The World Factbook -- World Career Education for a Global Economy BBC News Special Report -- Global Economy Guardian Special Report -- Global Economy World Bank Summary Trade Statistics for World ( hide ) Globalization Journals Outline Studies Aspects Alter - globalization Anti-globalization Counter-hegemonic globalization Cultural globalization Deglobalization Democratic globalization Economic globalization Environmental globalization Financial globalization Global citizenship education Global governance Global health History of globalization archaic early modern Military globalization Political globalization Trade globalization Workforce globalization Issues Global Disease Digital divide Labor arbitrage Population Warming Water crisis Other Brain drain reverse Climate change Climate justice Development aid Economic inequality Endangered languages Fair trade Forced migration Human rights Illicit financial flows Invasive species Investor - state disputes New international division of labour North -- South divide Offshoring Race to the bottom pollution havens Transnational crime McDonaldization Westernization American imperialism British Empire World war Theories Capital accumulation Dependency Development Earth system Fiscal localism Modernization ecological history of Primitive accumulation Social change World history World - systems Notable scholars Samir Amin Arjun Appadurai K. 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Question: what do the economic systems of the uk germany and russia have in common

reached an economy of at least US$2 trillion by GDP in nominal or PPP terms


Flag of Uruguay - Wikipedia Flag of Uruguay Jump to : navigation , search Uruguay Name The National Pavilion ( Official ) , The Sun and Stripes ( Nickname ) Use National flag and ensign Proportion 2 : 3 Adopted July 11 , 1830 Design Nine horizontal stripes of white alternate with blue with the white square on the upper hoist - side corner bearing the Sun of May in the center . Designed by Joaquín Suárez the first head of state of Uruguay in December 1828 and President of Uruguay 1843 - 1852 . The national flag of Uruguay ( Pabellón Nacional ) is one of the three official flags of Uruguay alongside with the Flag of Artigas and the Flag of the Treinta y Tres . It has a field of nine equal horizontal stripes alternating white and blue . The canton is white , charged with the Sun of May , from which 16 rays extend , alternating between triangular and wavy . The flag was first adopted by law on December 16 , 1828 , and had 19 stripes until July 11 , 1830 , when a new law reduced the number of stripes to nine . The flag was designed by Joaquín Suárez . Contents ( hide ) 1 Symbolism and design 2 Co-official flags 2.1 Maritime flag 3 Historical banners 3.1 Historical flags 3.2 Variations 4 Gallery 5 References 6 External links Symbolism and design ( edit ) The horizontal stripes on the flag represent the nine original departments of Uruguay , based on the U.S flag , where the stripes represent the original 13 colonies . The first flag designed in 1828 had 9 light blue stripes ; this number was reduced to 4 in 1830 due to visibility problems from distance . The Sun of May represents the May Revolution of 1810 ; according to the historian Diego Abad de Santillán , the Sun of May is a figurative sun that represents Inti , the sun god of the Inca religion . It also appears in the Flag of Argentina and the Coat of Arms of Bolivia . Co-official flags ( edit ) Flag of Artigas Flag of the Treinta y Tres Maritime flag ( edit ) Naval Jack of Uruguay Historical banners ( edit ) During Spanish rule : Flag of Viceroyalty of the Río de la Plata Flag of Cisplatina , under Brazilian occupation between 1821 and 1825 : Flag of the Cisplatina Historical flags ( edit ) Flag of the Provincia Oriental ( 1825 - 1828 ) Flag of the Oriental State of Uruguay ( 1828 - 1830 ) Variations ( edit ) During the Great Siege of Montevideo ( 1843 - 1851 ) Uruguay had two parallel governments , with two different flags : Flag used by the Government of the Cerrito during the Uruguayan Civil War . Flag used by the Government of the Defense during the Uruguayan Civil War . Gallery ( edit ) Flag in the historical Estadio Centenario , Montevideo . Flag in the Estadio Centenario Flag in Colonia del Sacramento Sun of May Flag in the Palacio Legislativo Historical Flag from the 1924 Olympic Games Flag in Otranto References ( edit ) Jump up ^ `` Flag of Uruguay '' . The World Factbook . Central Intelligence Agency . Archived from the original on June 13 , 2007 . Retrieved 2007 - 06 - 27 . ^ Jump up to : `` Uruguay , flag of '' . Guide to Hispanic Heritage . Britannica.com . Retrieved 2007 - 06 - 27 . External links ( edit ) Wikimedia Commons has media related to Flags of Uruguay . Uruguay at Flags of the World National Symbols of Uruguay National Flag Coat of Arms National Anthem Flag of Artigas Flag of the Treinta y Tres National Cockade Order of the Oriental Republic of Uruguay Medal of Military Merit ( Uruguay ) Order of Military Merit of the Companions of Artigas Flags of South America Sovereign states Argentina Bolivia Brazil Chile Colombia Ecuador Guyana Paraguay Peru Suriname Uruguay Venezuela Dependencies and other territories Falkland Islands French Guiana South Georgia and the South Sandwich Islands National flags and coats of arms National flags Sovereign states Dependent territories Timeline National coats of arms Sovereign states Dependent territories Heraldry portal This Uruguay - related article is a stub . You can help Wikipedia by expanding it . This flag - related article is a stub . You can help Wikipedia by expanding it . Retrieved from `` https://en.wikipedia.org/w/index.php?title=Flag_of_Uruguay&oldid=837762057 '' Categories : Flags of Uruguay National flags National symbols of Uruguay 1830 establishments in Uruguay Uruguay stubs Flag stubs Talk Contents About Wikipedia Afrikaans አማርኛ Azərbaycanca Беларуская Български Bosanski Català Čeština Dansk Deutsch Eesti Ελληνικά Español Esperanto Euskara فارسی Français Galego 한국어 Հայերեն हिन्दी Hrvatski বিষ্ণুপ্রিয়া মণিপুরী Bahasa Indonesia Italiano עברית ქართული Kiswahili Lietuvių Magyar Македонски मराठी Bahasa Melayu Nederlands 日本 語 Norsk Norsk nynorsk Polski Português Română Русский Scots Shqip Simple English Slovenčina Српски / srpski Srpskohrvatski / српскохрватски Suomi Svenska ไทย Türkçe Українська Tiếng Việt Yorùbá 中文 47 more Edit links This page was last edited on 22 April 2018 , at 22 : 00 . Text is available under the Creative Commons Attribution - ShareAlike License ; additional terms may apply . 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Question: what does the sun represent on the uruguay flag