Transreal Limits and Elementary Functions

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1 Transreal Limits an Elementary Functions Tiago S. os Reis, James A. D. W. Anerson Abstract We exten all elementary functions from the real to the transreal omain so that they are efine on ivision by zero. Our metho applies to a much wier class of functions so may be of general interest. Key wors: transreal continuity, transreal elementary functions, transreal limits, transreal numbers, transreal sequences, transreal series. 1 Introuction The set of real numbers, R, is extene to the set of transreal numbers, R T, by the aition of three, efinite, non-finite numbers: negative infinity, = 1/0; positive infinity, = 1/0; an nullity, Φ = 0/0. These three, non-finite numbers are calle strictly transreal numbers. Nullity was introuce in [1]. Transreal arithmetic is axiomatise an prove consistent by machine proof in [4]. A construction of the transreal numbers from the real numbers is given in [5]. That construction provies a human proof of the consistency of transreal arithmetic. Transreal limits are given in [3]. Transreal erivatives are given in [6]. Here we consier transreal limits an exten elementary functions of real numbers to transreal numbers. In [3] we efine a topology on R T which extens the topology of real numbers. Because of this we can efine, in a rigorous way, limits of sequences, series an functions an continuity of functions on R T so that wher- Tiago S. os Reis Feeral Institute of Eucation, Science an Technology of Rio e Janeiro; ; Brazil an Program of History of Science, Technique an Epistemology; Feeral University of Rio e Janeiro; ; Brazil; tiago.reis@ifrj.eu.br James A. D. W. Anerson School of Systems Engineering; University of Reaing; Whiteknights; Reaing; Englan; RG6 6AY; j.anerson@reaing.ac.uk 1

2 Tiago S. os Reis, James A. D. W. Anerson ever real numbers occur in real limits, they occur ientically in transreal limits an wherever infinities occur as symbols in extene real limits, they occur ientically in transreal limits but as efinite numbers. This means that transreal topology agrees with the usual, real topology. We exten every elementary function to the transreal omain. Some of these functions, such as the exponential an trigonometric functions, were efine, in [], at strictly transreal numbers, motive by power series. However the convergence of series in [] is taken in an intuitive way, without any rigorous efinition of the limit of sequences an series on R T. That evelopment also lacks a topology on R T. Here we also use power series but first we extene some results about series to R T using its well-efine topology [3]. Topological Founations In [3] we efine a topology, limit of sequences an limit an continuity of functions on R T. Below we summarise this content. Definition 1 A set is open on R T if an only if it is compose of arbitrarily many unions of finitely many intersections of the following four kins of interval: i) (a,b) where a,b R, ii) [,b) where b R, iii)(a, ] where a R an iv) {Φ}. The reaer can verify that these open sets o, in fact, make a topology on R T. Proposition 1 R T is a Hausorff, isconnecte, separable, compact an completely metrisable space. For efinitions of these terms an proofs see [3] an a forthcoming paper [7]. Notice that Φ is the unique isolate point of R T. Proposition The topology on R, inuce by the topology of R T, is the usual topology of R. That is if A R T is open on R T then A R is open (in the usual sense) on R an if A R is open (in the usual sense) on R then A is open on R T. Remember the efinition of a sequence. A sequence in R T is a function x : N R T. We customarily write x n in place of x(n) an write (x n ) n N in place of x : N R T. We use the usual efinition for the convergence of a sequence in a topological space. That is a sequence, (x n ) n N R T, converges to x R T if an only if for each neighbourhoo, V R T of x, there is n V N such that x n V for all n n V. Notice that since R T is a Hausorff space, the limit of a sequence, when it exists, is unique. Remark 1 Let (x n ) n N R an let L R. Notice that lim n x n = L in R T if an only if lim n x n = L in the usual sense in R. Furthermore, (x n ) n N iverges, in

3 Transreal Limits an Elementary Functions 3 the usual sense, to negative infinity if an only if lim n x n = in R T. Similarly (x n ) n N iverges, in the usual sense, to infinity if an only if lim n x n = in R T. Remark Let (x n ) n N R T. Notice that lim n x n = Φ if an only if there is k N such that x n = Φ for all n k. Proposition 3 Every monotone sequence of transreal numbers is convergent. Proposition 4 (Transreal version of the Bolzano-Weierstrass Theorem) Every sequence of transreal numbers has a convergent subsequence. 3 Transreal series In this section we exten some results on series to the transreal omain. Definition Let (x n ) n N R T. For each n N, we efine s n := n i=1 x i = x x n. The sequence (s n ) n N is calle a series an is enote by x n, each s n is calle a partial sum of x n an x n is calle the n-th term of x n. We say that x n converges or is convergent if an only if there is the lim n s n. Otherwise, x n iverges or is ivergent. When x n is convergent we enote x n := lim s n. n n=1 Definition 3 Let (x n ) n N R T. We say that a series x n converges absolutely or is absolutely convergent if an only if x n is convergent. Customarily, in a calculus course, the first example of a convergent series is the geometric series, r n. Recall that r n converges in R if an only if r ( 1,1) an, in this case, n=1 rn = r 1 r. Example 1 Let r R T. The series r n converges in R T if an only if r {,Φ} ( 1, ]. Inee, if r {,Φ} then n=1 rn = Φ, if r ( 1,1) then n=1 rn = 1 r r, if r [1, ] then n=1 rn = an if r (, 1] then s n 1 < 0 an s n 0 for all n N whence r n iverges. Remark 3 In the real omain we know that if x n is convergent then lim n x n = 0. In the transreal omain there is no similar result. That is, we can have a convergent series, x n, such that (x n ) n N converges to anywhere in R T. Inee, if a {Φ,, } an x n = a for all n N then x n is convergent (to a) an lim n x n = a. If a R is positive, let x n = a 1 for each n N. Remember that n

4 4 Tiago S. os Reis, James A. D. W. Anerson 1 converges increasingly to π n 6 whence n i=1 1 < π i 6 for all n N. Given an arbitrary positive M R, there is n M N such that n M a > M + π 6. Hence if n n M then na n M a > M + π 6 > M + n i=1 1 whence i n i=1 x n = na n i=1 1 > M, that i is, n i=1 x n (M, ]. Thus x n is convergent (to ) an lim n x n = a. If a R is negative, let x n = a + 1 for each n N then, by a similar argument, we have that n x n is convergent (to ) an lim n x n = a. Remark 4 In the real omain, we know that a series x n is convergent if an only if for each positive ε R there is n ε N such that n x i < ε (1) i=m whenever n m n ε. This is nothing more than the application of the Cauchy criterion. As R is a complete metric space, a sequence is convergent if an only if it is Cauchy. The inequality (1) is, in fact, s n s m 1 < ε. We can naturally enunciate that result in the transreal omain by using its complete metric, enote as. A series x n is convergent if an only if for each positive ε R there is n ε N such that (s n,s m 1 ) < ε whenever n m n ε. This is correct but is not as helpful as in the real omain because, unlike in real omain, the fact that (s n,s m 1 ) is less than an arbitrary, positive, real number is not equivalent to the fact that n i=m x i is less than some positive, real number. We can enunciate the Cauchy criterion, in the real omain, in another way. A series x n is convergent if an only if for each neighbourhoo V of zero there is n V N such that n i=m x i V whenever n m n V. In this way we might imagine that there is a similar result for the transreal omain. For example, let x n = n for all n N. The series x n is convergent (to ) an for each neighbourhoo V of infinity there is n V N such that n i=m x i V whenever n m n V. However this is not always the case. Let x n = 1 for all n N. The series x n is convergent (to ), but for all n N, n i=n x i = 1 oes not belong to neighbourhoo of infinity (M, ] where M > 1. We might wish to enunciate, in the transreal omain, something like: if the series x n is convergent then either n i=m x i belongs to a neighbourhoo of zero for n an m large enough or n i=m x i belongs to a neighbourhoo of infinity for n an m large enough or else (x n ) n N is constant. But this is still not true. Let x n = n 1. The series x n is convergent (to ) but oes not satisfy any of the three conitions above. In fact, let there be arbitrary, positive ε,m R such that ε < M. There is n N such that n i=n x i = n 1 ( ε,ε) an there is m > n such that m i=n x i (M, ]. Remark 5 In the real omain, we know that if x n is absolutely convergent then x n is convergent. In the transreal omain this is not true. Inee, accoring to Example 1, the series ( 1) n is absolutely convergent ( ( 1) n converges to ), but it is not convergent. Proposition 5 In the transreal omain every series of non-negative terms is convergent. In other wors, if (x n ) n N R T an x n 0 for all n N then x n is convergent.

5 Transreal Limits an Elementary Functions 5 Proof. As x n 0 for all n N, for some inex n 0, the sequence of partial sums of x n+n0, (s n+n0 ) n N is monotone. By Proposition 3, x n+n0 is convergent whence x n is convergent. Obviously if (x n ) n N R T an x n 0 for all n N then x n is convergent too. That is, every series of non-positive terms is convergent. Remark 6 Notice that, since in the transreal omain every series of non-negative terms is convergent, the Comparison Test, the Ratio Test an the Root Test are not appropriate for transreal series. Definition 4 Let (c n ) n N R T. The series c 0 + c n x n is calle a power series. A power series efines a function at the values x for which it is convergent. That is, if A is the set of all transreal numbers x such that c 0 + c n x n converges then is a well-efine function. f : A R T x c 0 + n=1 c nx n 4 Elementary functions Recall that a real, elementary function is efine in the following way. Every polynomial, root, exponential, logarithmic, trigonometric an inverse trigonometric function is an elementary function; any finite composition of elementary functions is an elementary function; an any finite combination, using the four arithmetical operations, between elementary functions is an elementary function. We wish to exten the real elementary functions to the transreal numbers. There are some choices which nee to be mae. The power series of some elementary functions have a value, at a strictly transreal number, that is ifferent to the limit of the function when the argument tens to that strictly transreal number. Let us illustrate this with the exponential function. It is known that e x = 1+ n=1 xn n! for all x R. We have that lim x e x = 0 but 1 + ( ) n n=1 n! = Φ. How shoul we efine e? By e = 0 or e = Φ? Geometrically speaking, it is intuitive to us that, in the graph of the exponential function, when x hits minus infinity, e x hits zero. In this way we choose to efine e = 0. This has the avantage that the exponential function becomes continuous at. At there is no trouble because lim x e x = = 1 + n=1 n n!. Nullity, Φ, is an isolate point, whence it is nonsense to speak of the limit at Φ; because of this we choose to efine e Φ by way of the power series. We have that 1 + n=1 Φn n! = Φ whence we efine e Φ = Φ. However we o not always efine a function by way of limits or power series. For example the function x, calculate exactly at zero is sin(0) 0 = 0 0 an 0 0 = Φ but lim x 0 x = 1. Again we have a choice to make. The transreal numbers were efine in orer to allow ivision by zero. It seems to us counter-prouctive to ignore

6 6 Tiago S. os Reis, James A. D. W. Anerson the fact that, in R T, 0 sin(0) 0 is well-efine. Hence we choose to efine 0 = Φ. Of course the function x becomes iscontinuous at 0 but this is the price to pay for having transreal arithmetic. In summary we aopt the following proceure to exten a function from the real omain to the transreal omain. If the expression of the function is lexically wellefine, at a transreal number, then we efine the function by simply applying its expression at that transreal number. If the function f is not lexically well-efine at a transreal number, x 0, but there is a limit, lim x x0 f (x), then we choose to efine the function at x 0 by lim x x0 f (x). Otherwise we choose to efine the function by way of its power series if it converges. An if, nevertheless, its power series oes not converge, we keep the function unefine. Firstly we recall some results about the limit an continuity of functions obtaine in [3]. Recall that if X is a topological space then x 0 A X is a limit point of A if an only if for every neighbourhoo V of x 0 it follows that V (A \ {x 0 }) = /0. The set of all limit points of A is enote as A. We use the usual efinition of the limit of functions in a topological space. That is if A is a subset of R T, f : A R T is a function, x 0 is a limit point of A an L is a transreal number, we say that lim x x0 f (x) = L if an only if, for each neighbourhoo V of L, there is a neighbourhoo U of x 0 such that f (A U \ {x 0 }) V. Remark 7 Notice that given x 0,L R, lim x x0 f (x) = L, in the transreal sense, if an only if lim x x0 f (x) = L, in the real sense. The same can be sai about lim x x0 f (x) =, lim x x0 f (x) =, lim x f (x) = L, lim x f (x) =, lim x f (x) =, lim x f (x) = L, lim x f (x) = an lim x f (x) =. Remark 8 Let x 0 R T, notice that lim x x0 f (x) = Φ if an only if there is a neighbourhoo U of x 0 such that f (x) = Φ for all x U \ {x 0 }. We use the usual efinition of continuity in a topological space. That is if A R T, f : A R T is a function an x 0 A, we say that f is continuous in x 0 if an only if, for each neighbourhoo V of f (x 0 ), there is a neighbourhoo U of x 0 such that f (A U) V. We say that f is continuous in A if an only if f is continuous in x for all x A. Remark 9 Of course if x 0 is a limit point of A then f is continuous in x 0 if an only if lim x x0 f (x) = f (x 0 ). Remark 10 Notice that given x 0 R, f is continuous in x 0, in the transreal sense, if an only if f is continuous in x 0, in the real sense. Remark 11 Notice that if Φ belongs to the omain of f then f is continuous in Φ. 4.1 Polynomial Functions A function, f, is a real, polynomial function if an only if there is n N an a 0,...,a n R such that f (x) = a n x n + + a 1 x + a 0 for all x R. Of course, f

7 Transreal Limits an Elementary Functions 7 can be written f (x) = a m x m + + a 1 x + a 0 for any m > n provie that a n+1 = = a m = 0. An if a k = 0 for some k {1,...,n} then f can also be written f (x) = a n x n + + a k+1 x k+1 + a k 1 x k a 1 x + a 0. This multiplicity, in the representation of f, is a problem in the transreal omain. In the real omain, 0 x k = 0 for all real x but 0 x k = 0 oes not hol for all transreal x. We have 0 ( ) k = 0 k = 0 Φ k = Φ 0. In orer to avoi this problem we efine that a function f is a real, polynomial function if an only if there is n,k N, n 1,...,n k {1,...,n 1} an a 0,a n1...,a nk,a n R such that a n1...,a nk,a n are ifferent from zero an f : R R x a n x n + a nk x n k + + a n1 x n 1 + a 0. () By following our proceure, as every arithmetical operation is well-efine in transreal numbers, we exten the function f to R T naturally in the following way. Definition 5 A function f is a transreal, polynomial function if an only if there is n,k N, n 1,...,n k {1,...,n 1} an a 0,a n1...,a nk,a n R such that a n1...,a nk,a n are ifferent from zero an f : R T R T x a n x n + a nk x n k + + a n1 x n 1 + a 0. Remark 1 For every non-constant, transreal, polynomial function, f, we have that f (Φ) = Φ. Notice that, in the transreal omain, some polynomial functions are not continuous. Example Let n N such that n an let non-zero numbers a 0,...,a n R such that a n a n 1 < 0 an f (x) = a n x n + +a 1 x +a 0 then f ( ) = Φ but lim x f (x) = sgn(a n ), whence f is not continuous at. Example 3 Let n N such that n an let non-zero numbers a 0,...,a n R such that a n a n 1 > 0 an f (x) = a n x n + + a 1 x + a 0 then f ( ) = Φ but lim x f (x) = sgn(a n ) ( ) n, whence f is not continuous at. If the reaer wishes to exten the function in () continuously to R T then bounary conitions are neee at an : f : R T R{ T an x x n, if x {, }. a n x n + a nk x n k + + a n1 x n 1 + a 0, otherwise Notice that just by using Definition 5, some particular, transreal, polynomial functions become continuous as exemplifie next. Example 4 Let a,b R such that a 0 an f (x) = ax + b for all x R T. Notice that lim x f (x) = f ( ) an lim x f (x) = f ( ) whence f is continuous in R T.

8 8 Tiago S. os Reis, James A. D. W. Anerson 4. Exponential Functions A function, f, is a real, exponential function if an only if there is a positive a R such that a 1 an f : R R x a x. (3) The natural, exponential function is f (x) = e x where e is Euler s number. We know that every exponential function f (x) = a x can be written f (x) = e ln(a)x where lna is the natural logarithm of a. Let f : R R x e x. By following our proceure, as the function f is not efine by way of arithmetical operations, we efine f ( ) := lim x e x an f ( ) := lim x f (x) whence f ( ) = 0 an f ( ) =. Furthermore, we know that e x = 1+ n=1 xn n! for all x R. Following our metho, in the absence of limit, we efine f (Φ) = 1 + n=1 Φn n!, whence f (Φ) = Φ. Thus we exten f to R T by efining f : R T R T { 0, if x = x e x. = 1 + n=1 xn n!, otherwise Now we exten the function in (3) to R T in the following way. Definition 6 A function, f, is a transreal, exponential function if an only if there is a positive a R such that a 1 an f : R T R T x a x = e ln(a)x. Proposition 6 (Properties of exponential functions) Let f be the transreal, exponential function, f (x) = a x. We have that, for all x,y R T : 1. f is continuous.. The image of f is [0, ] {Φ}. 3. f is strictly monotone. 4. f is injective. 5. a x+y = a x a y. 6. a x y = ax a y. Remark 13 In [6] we efine the transreal erivative in the following way. Let A R T, f : A R T an x 0 A. If x 0 R A an f is ifferentiable at x 0, in the real sense, then the transreal erivative of f at x 0 is the real erivative f (x 0 ). If x 0 {, } D (where D enotes the set of points in A at which f is ifferentiable, in the real sense) then the transreal erivative of f at x 0 is f (x 0 ) = lim x x0 f (x) if this limit exist. An if x 0 / A then the transreal erivative of f at x 0 is f (x 0 ) := Φ.

9 Transreal Limits an Elementary Functions 9 Remark 14 Accoring to Remark 13, we have that x a x = ln(a)a x for all x R T. In particular, x e x = e x for all x R T. 4.3 Logarithmic Functions A function, f, is a real, logarithmic function if an only if there is a positive a R such that a 1 an f : (0, ) R (4) x log a (x) where log a (x) enotes the unique, real number y, for which a y = x. By following our proceure, as the transreal exponential is injective an has image [0, ] {Φ}, the function f in (4) is lexically well-efine in [0, ] {Φ}. Because of this we exten the function f to R T in the following way. Definition 7 A function, f, is a transreal, logarithmic function if an only if there is a positive a R such that a 1 an f : [0, ] {Φ} R T x log a (x) where log a (x) enotes the unique, transreal number y, for which a y = x. Like in the real omain, we call log e (x) the transreal, natural logarithm of x an enote it as ln(x). In particular we have that ln(0) =, ln( ) = an ln(φ) = Φ. Proposition 7 (Properties of logarithmic functions) Let f be the transreal, logarithmic function, f (x) = log a (x). We have that for all x,y R T : 1. f is continuous.. The image of f is R T. 3. f is strictly monotone. 4. f is injective. 5. log a (x y) = log a (x) + log a (y). 6. log a ( xy ) = log a (x) log a (y). Remark 15 We have alreay sai that the erivative of a function f at, if D, is f ( ) := lim x f (x) if this limit exists. Fortunately this efinition has sense for all x 0 D so we efine, for all x 0 D, f (x 0 ) := lim x x0 f (x), if this limit exists. Remark 16 Accoring to Remark 15, x log a (x) 0 = lim x 0 1 ln(a) 1 x = lim x ln(a) { 1 x = 1, if 0 < a < 1 ln(a) =. Therefore we have that, if a > 1 x log a (x) = 1 ln(a) 1 x for all x R T. In particular, x ln(x) = 1 x for all x RT.

10 10 Tiago S. os Reis, James A. D. W. Anerson Remark 17 The Definition 6 efines powers x y when x R is positive an x 1. The transreal logarithm allows us to efine x y for all non-negative transreal x. Let x [0, ] {Φ} an y R T, we efine x y := e ln(x)y. In particular, a) 0 y = for all y < 0. b) 0 0 = Φ. c) 0 y = 0 for all y > 0. ) 0 Φ = Φ. e) 1 y = 1 for all y R. f) 1 y = Φ for all y {,,Φ}. g) y = 0 for all y < 0. h) 0 = Φ. i) y = for all y > 0. j) Φ = Φ. k) Φ Φ = Φ. 4.4 Trigonometric functions We know that the real, trigonometric functions can be efine as follows. a) b) c) ) e) f) sin : R R ( 1) n+1, (n 1)! xn 1 x = n=1 cos : R R x cos(x) = 1 + ( 1) n n=1 (n)! xn, tan : R \ { π + kπ; k Z } R x tan(x) =, cos(x) csc : R \ {kπ; k Z} R x csc(x) = 1, sec : R \ { π + kπ; k Z } R x sec(x) = 1 cos(x) cot : R \ {kπ; k Z} R x cot(x) = cos(x). If f is a trigonometric function, that is, if f is one of the six functions above then there are no limits lim x f (x) an lim x f (x). So by following our proceure, we efine sin( ) := ( 1) n+1 n=1 (n 1)! ( )n 1, cos( ) := 1 + ( 1) n n=1 (n)! ( )n, tan( ) := sin( ) 1 1 cos( ), csc( ) := sin( ), sec( ) := cos( ) an cot( ) :=. An similarly for an Φ. Hence sin( ) = cos( ) = tan( ) = cos( ) sin( ) csc( ) = sec( ) = cot( ) = sin( ) = cos( ) = tan( ) = csc( ) = sec( ) = cot( ) = sin(φ) = cos(φ) = tan(φ) = csc(φ) = sec(φ) = cot(φ) = Φ. an

11 Transreal Limits an Elementary Functions 11 Notice that tan, csc, sec an cot are lexically well-efine at π + kπ an kπ for all k Z. In this way we exten the trigonometric functions to R T in the following way. Definition 8 A function is a transreal, trigonometric function if an only if it is one of the functions below. a) sin : R T R T x = n=1 b) cos : R T R T x cos(x) = 1 + n=1 c) tan : R T R T x tan(x) =, cos(x) ) e) f) csc : R T R T x csc(x) = 1, sec : R T R T x sec(x) = 1 cos(x) cot : R T R T x cot(x) = cos(x) ( 1) n+1, (n 1)! xn 1. an ( 1) n, (n)! xn Proposition 8 (Properties of trigonometric functions) 1. Every transreal, trigonometric function is iscontinuous.. The image of sin an cos is [ 1,1] {Φ}. 3. The image of tan an cot is R T. 4. The image of csc an sec is (, 1] [1, ] {Φ}. 5. Every transreal, trigonometric function has perio π. 6. sin restricte to [ π,π] {Φ} is injective. 7. cos restricte to [0,π] {Φ} is injective. 8. tan restricte to [ π, π ] {Φ} is injective. 9. csc restricte to [ π, π ] {Φ} is injective. 10. sec restricte to [0,π] {Φ} is injective. 11. cot restricte to [0,π] {Φ} is injective. 1. sin (x) + cos (x) = 1 x for all x R T. 13. tan (x) + 1 x = sec (x) for all x R T. 14. cot (x) + 1 x = csc (x) for all x R T. Remark 18 We have alreay efine the erivative of a function f at x 0, if x 0 D, by f (x 0 ) := lim x x0 f (x) if this limit exists. Regrettably if f is a trigonometric function an x 0 {, } then there is no limit lim x x0 f (x). If we appeal to lexical application of the erivative, we can aopt the following. If x 0 D an there is no limit lim x x0 f (x) an the expression of f is lexically well-efine at x 0 then we efine the erivative of f at x 0 as f (x 0 ). Remark 19 For example x = cos(x) for all x R, an are limit points of the set where sin is ifferentiable, in the real sense, an there are no

12 1 Tiago S. os Reis, James A. D. W. Anerson lim x cos(x) an lim x cos(x). As cos(x) is lexically well-efine for x = an x =, accoring to Remark 18, x = cos( ) an x = cos( ). An since Φ is not a limit point of the omain of sin, accoring to Remark 13, x Φ = Φ = cos(φ). Because of this, x = cos(x) for all x R T. As another example x tanx = sec (x) for all x R \ { π + kπ; k Z }. As, for every k Z, π + kπ is a limit point of the set where tan is ifferentiable, in the real sense, an there is lim x π +kπ sec (x), accoring to Remark 15, we efine x tan(x) π +kπ = lim x π +kπ sec (x) = = sec ( π + kπ ). Since an are limit points of the set where tan is ifferentiable, in the real sense, an there are no limits lim x sec (x) an lim x sec (x) but sec (x) is lexically well-efine for x = an x = so, accoring to Remark 18, x tan(x) = sec ( ) an x tan(x) = sec ( ). An since Φ is not a limit point of the omain of tan, accoring to Remark 13, x tan(x) Φ = Φ = sec (Φ). Because of this, x tan(x) = sec (x) for all x R T. Generally, we have, for all x R T : 1. x = cos(x).. x cos(x) = cos(x). 3. x tan(x) = sec (x). 4. x csc(x) = csc(x)cot(x). 5. x sec(x) = sec(x)tan(x). 6. x cot(x) = csc (x). At this point the reaer can alreay efine the inverse trigonometric functions an euce their properties. 5 The limit an value of a function at a point We tabulate both the values an the limits of the above functions so that we can see whether a function is extene to R T continuously or not. a x + a 1 x where a > 0 an a 1 > 0 x 0 a x + a 1 x lim x x0 (a x + a 1 x) Continuity at x 0 Φ not continuous continuous a x + a 1 x where a > 0 an a 1 < 0 x 0 a x + a 1 x lim x x0 (a x + a 1 x) Continuity at x 0 continuous Φ not continuous

13 Transreal Limits an Elementary Functions 13 e x x 0 e x 0 lim x x0 e x Continuity at x continuous continuous ln(x) x 0 ln(x 0 ) lim x x0 ln(x) Continuity at x continuous continuous x 0 sin(x 0 ) lim x x0 Continuity at x 0 - Φ none not continuous Φ none not continuous cos(x) x 0 cos(x 0 ) lim x x0 cos(x) Continuity at x 0 - Φ none not continuous Φ none not continuous tan(x) x 0 tan(x 0 ) lim x x0 tan(x) Continuity at x 0 Φ none not continuous Φ none not continuous π + (k + 1)π none not continuous π + kπ none not continuous tan(x) restricte to [ π, π ] x 0 tan(x 0 ) lim x x0 tan(x) Continuity at x 0 π - - continuous π continuous

14 14 Tiago S. os Reis, James A. D. W. Anerson csc(x) x 0 csc(x 0 ) lim x x0 csc(x) Continuity at x 0 - Φ none not continuous Φ none not continuous kπ none not continuous csc(x) restricte to [ π, π ] x 0 csc(x 0 ) lim x x0 csc(x) Continuity at x 0 0 none not continuous sec(x) x 0 sec(x 0 ) lim x x0 sec(x) Continuity at x 0 - Φ none not continuous Φ none not continuous π + kπ none not continuous sec(x) restricte to [0,π] x 0 sec(x 0 ) lim x x0 sec(x) Continuity at x 0 π none not continuous cot(x) x 0 cot(x 0 ) lim x x0 cot(x) Continuity at x 0 - Φ none not continuous Φ none not continuous kπ none not continuous (k + 1)π - none not continuous cot(x) restricte to [0,π] x 0 cot(x 0 ) lim x x0 cot(x) Continuity at x 0 0 continuous π - - continuous x 0 sin(x 0 ) x 0 x lim x x0 x Continuity at x 0 0 Φ 1 not continuous - Φ 0 not continuous Φ 0 not continuous

15 Transreal Limits an Elementary Functions 15 6 Conclsion We exten all elementary functions from the real omain to the transreal omain so that they are efine on ivision by zero. If the expression of the function is lexically well-efine, at a transreal number, then we efine the function by simply applying its expression at that transreal number. If the function f is not lexically well-efine at a transreal number, x 0, but there is a limit, lim x x0 f (x), then we choose to efine the function at x 0 by lim x x0 f (x). Otherwise we choose to efine the function by way of its power series if it converges. An if, nevertheless, its power series oes not converge, we keep the function unefine. This metho for extening functions from the real omain to the transreal omain clearly works for a much wier class of functions so it may be of general interest. Acknowlegements The authors woul like to thank the members of Transmathematica for many helpful iscussions. The first author s research was financially supporte, in part, by the Feeral Institute of Eucation, Science an Technology of Rio e Janeiro, campus Volta Reona an by the Program of History of Science, Technique an Epistemology, Feeral University of Rio e Janeiro. The secon author s research was financially supporte, in part, by the School of Systems Engineering at the University of Reaing an by a Research Travel Grant from the University of Reaing Research Enowment Trust Fun (RETF). References 1. J. A. D. W. Anerson. Representing geometrical knowlege. Phil. Trans. Roy. Soc. Lon. Series B., 35(1358): , James A. D. W. Anerson. Perspex machine ix: Transreal analysis. In Longin Jan Lateki, Davi M. Mount, an Angela Y. Wu, eitors, Vision Geometry XV, volume 6499 of Proceeings of SPIE, pages J1 J1, James A. D. W. Anerson an Tiago S. os Reis. Transreal limits expose category errors in ieee 754 floating-point arithmetic an in mathematics. In Lecture Notes in Engineering an Computer Science: Proceeings of The Worl Congress on Engineering an Computer Science 014, WCECS 014, -4 October, 014, San Francisco, USA., volume 1, pages 86 91, James A. D. W. Anerson, Norbert Völker, an Anrew A. Aams. Perspex machine viii: Axioms of transreal arithmetic. In Longin Jan Lateki, Davi M. Mount, an Angela Y. Wu, eitors, Vision Geometry XV, volume 6499 of Proceeings of SPIE, pages.1.1, Tiago S. os Reis an James A. D. W. Anerson. Construction of the transcomplex numbers from the complex numbers. In Lecture Notes in Engineering an Computer Science: Proceeings of The Worl Congress on Engineering an Computer Science 014, WCECS 014, -4 October, 014, San Francisco, USA., volume 1, pages 97 10, Tiago S. os Reis an James A. D. W. Anerson. Transifferential an transintegral calculus. In Lecture Notes in Engineering an Computer Science: Proceeings of The Worl Congress on Engineering an Computer Science 014, WCECS 014, -4 October, 014, San Francisco, USA., volume 1, pages 9 96, Tiago S. os Reis an James A. D. W. Anerson. Transreal calculus. IAENG International Journal of Applie Mathematics, 45(1):51 63, 015.

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