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But since (x1 < x2 ) ⇔ (a x1 < a x2 ) (when a > 1), for any numbers x1 , x2 ∈ R such that x1 ∈ A and x2 ∈ B we have x1 < x2 . Consequently, theaxiom of completeness is applicable to the sets A and B, and it follows that thereexists x0 such that x1 ≤ x0 ≤ x2 for all x1 ∈ A and x2 ∈ B. We shall show thata x0 = y0 .1223LimitsIf a x0 were less than y0 , then, since a x0 +1/n → a x0 as n → ∞, there would be anumber n ∈ N such that a x0 +1/n < y0 . Then we would have (x0 + n1 ) ∈ A, while thepoint x0 separates A and B.
Hence the assumption a x0 < y0 is untenable. Similarlywe can verify that the inequality a x0 > y0 is also impossible. By the properties ofreal numbers, we conclude from this that a x0 = y0 .140 We have assumed up to now that a > 1. But all the constructions could berepeated for 0 < a < 1. Under this condition 0 < a r < 1 if r > 0, so that in 60 and100 we now find that (x1 < x2 ) ⇒ (a x1 > a x2 ) where 0 < a < 1.Thus for a > 0, a = 1, we have constructed a real-valued function x → a x on theset R of real numbers with the following properties:1) a 1 = a;2) a x1 · a x2 = a x1 +x2 ;3) a x → a x0 as x → x0 ;4) (a x1 < a x2 ) ⇔ (x1 < x2 ) if a > 1, and (a x1 > a x2 ) ⇔ (x1 < x2 ) if 0 < a < 1;5) the range of values of the mapping x → a x is R+ = {y ∈ R | 0 < y}, the setof positive numbers.Definition 7 The mapping x → a x is called the exponential function with base a.The mapping x → ex , which is the case a = e, is encountered particularly oftenand is frequently denoted exp x.
In this connection, to denote the mapping x → a x ,we sometimes also use the notation expa x.b) The logarithmic function. The properties of the exponential function show thatit is a bijective mapping expa : R → R+ . Hence it has an inverse.Definition 8 The mapping inverse to expa : R → R+ is called the logarithm to basea (0 < a, a = 1), and is denotedloga : R+ → R.Definition 9 For base a = e, the logarithm is called the natural logarithm and isdenoted ln : R+ → R.The reason for the terminology becomes clear under a different approach to logarithms, one that is in many ways more natural and transparent, which we shallexplain after constructing the fundamentals of differential and integral calculus.By definition of the logarithm as the function inverse to the exponential function,we have ∀x ∈ R loga a x = x ,∀y ∈ R+ a loga y = y .It follows from this definition and the properties of the exponential function inparticular that in its domain of definition R+ the logarithm has the following properties:3.2 The Limit of a Function1231 ) loga a = 1;2 ) loga (y1 · y2 ) = loga y1 + loga y2 ;3 ) loga y → loga y0 as R+ y → y0 ∈ R+ ;4 ) (loga y1 < loga y2 ) ⇔ (y1 < y2 ) if a > 1 and (loga y1 > loga y2 ) ⇔ (y1 < y2 )if 0 < a < 1;5 ) the range of values of the function loga : R+ → R is the set R of all real numbers.Proof We obtain 1 ) from property 1) of the exponential function and the definitionof the logarithm.We obtain property 2 ) from property 2) of the exponential function.
Indeed, letx1 = loga y1 and x2 = loga y2 . Then y1 = a x1 and y2 = a x2 , and so by 2), y1 · y2 =a x1 · a x2 = a x1 +x2 , from which it follows that loga (y1 · y2 ) = x1 + x2 .Similarly, property 4) of the exponential function implies property 4 ) of the logarithm.It is obvious that 5) ⇒ 5 ).Property 3 ) remains to be proved.By property 2 ) of the logarithm we haveloga y − loga y0 = logay,y0and therefore the inequalities−ε < loga y − loga y0 < εare equivalent to the relationloga a −ε = −ε < logayy0 < ε = loga a ε ,which by property 4 ) of the logarithm is equivalent toy< a ε for a > 1,y0yaε << a −ε for 0 < a < 1.y0−a ε <In any case we find that ify0 a −ε < y < y0 a εwhen a > 1ory0 a ε < y < y0 a −εwhen 0 < a < 1,we have−ε < loga y − loga y0 < ε.1243LimitsFig.
3.2Thus we have proved thatlimR+ y→y0 ∈R+loga y = loga y0 .Figure 3.2 shows the graphs of the functions ex , 10x , ln x, and log10 x =: log x;Fig. 3.3 gives the graphs of ( 1e )x , 0.1x , log1/e x, and log0.1 x.We now give a more detailed discussion of one property of the logarithm that weshall have frequent occasion to use.We shall show that the equality6 ) loga bα = α loga bholds for any b > 0 and any α ∈ R.Proof 10 The equality is true for α = n ∈ N. For by property 2 ) of the logarithmand induction we find loga (y1 · · · yn ) = loga y1 + · · · + loga yn , so that loga bn = loga b + · · · + loga b = n loga b.20 loga (b−1 ) = − loga b, for if β = loga b, thenb = aβ ,b−1 = a −βandloga b−1 = −β.30 From 10 and 20 we now conclude that the equality loga (bα ) = α loga b holdsfor α ∈ Z.40 loga (b1/n ) = n1 loga b for n ∈ Z. Indeed,nloga b = loga b1/n = n loga b1/n .3.2 The Limit of a Function125Fig.
3.350 We can now verify that the assertion holds for any rational number α =In fact,mmloga b = m loga b1/n = loga b1/n = loga bm/n .nmn∈ Q.60 But if the equality loga br = r loga b holds for all r ∈ Q, then letting r inQ tend to α, we find by property 3) for the exponential function and 3 ) for thelogarithm that if r is sufficiently close to α, then br is close to bα and loga br isclose to loga bα .
This means thatlim loga br = loga bα .Qr→αBut loga br = r loga b, and thereforeloga bα = lim loga br = lim r loga b = α loga b.Qr→αQr→αFrom the property of the logarithm just proved, one can conclude that the following equality holds for any α, β ∈ R and a > 0:6) (a α )β = a αβ .Proof For a = 1 we have 1α = 1 by definition for all α ∈ R. Thus the equality istrivial in this case.If a = 1, then by what has just been proved we have β = β loga a α = β · α loga a = β · α = loga a αβ ,loga a αwhich by property 4 ) of the logarithm is equivalent to this equality.c) The power function.
If we take 1α = 1, then for all x > 0 and α ∈ R we havedefined the quantity x α (read “x to power α”).1263LimitsFig. 3.4Definition 10 The function x → x α defined on the set R+ of positive numbers iscalled a power function, and the number α is called its exponent.A power function is obviously the composition of an exponential function andthe logarithm; more preciselyx α = a loga (xα)= a α loga x .Figure 3.4 shows the graphs of the function y = x α for different values of theexponent.3.2.3 The General Definition of the Limit of a Function (Limitover a Base)When proving the properties of the limit of a function, we verified that the onlyrequirements imposed on the deleted neighborhoods in which our functions weredefined and which arose in the course of the proofs were the properties B1 ) andB2 ), mentioned in the introduction to the previous subsection.
This fact justifies thedefinition of the following mathematical object.a. Bases; Definition and Elementary PropertiesDefinition 11 A set B of subsets B ⊂ X of a set X is called a base in X if thefollowing conditions hold:B1 ) ∀B ∈ B (B = ∅);B2 ) ∀B1 ∈ B ∀B2 ∈ B ∃B ∈ B (B ⊂ B1 ∩ B2 ).In other words, the elements of the collection B are nonempty subsets of X andthe intersection of any two of them always contains an element of the same collection.In Table 3.1 we list some of the more useful bases in analysis.3.2 The Limit of a Function127Table 3.1Notation forthe baseReadSets (elements) of thebaseDefinition of and notationfor elementsx→ax tends to aDeleted neighborhoodsof a ∈ RŮ (a) := {x ∈ R | a − δ1 << x < a + δ2 ∧ x = a},where δ1 > 0, δ2 > 0x→∞x tends to infinityNeighborhoods ofinfinityU (∞) := {x ∈ R | δ < |x|},where δ ∈ Rx → a, x ∈ EorEx→aorx −→∈E ax tends to a in EDeleted neighborhoods*of a in EŮ E (a) := E ∩ Ů (a)x → ∞, x ∈ EorEx→∞orx −→∈E ∞x tends toinfinity in ENeighborhoods** ofinfinity in EUE (∞) := E ∩ U (∞)* Itis assumed that a is a limit point of E** Itis assumed that E is not boundedIf E = Ea+ = {x ∈ R | x > a} (resp.
E = Ea− = {x ∈ R | x < a}) we writex → a + 0 (resp. x → a − 0) instead of x → a, x ∈ E, and we say that x tendsto a from the right (resp. x tends to a from the left) or through larger values(resp. through smaller values). When a = 0 it is customary to write x → +0 (resp.x → −0) instead of x → 0 + 0 (resp. x → 0 − 0).The notation E x → a + 0 (resp. E x → a − 0) will be used instead ofx → a, x ∈ E ∩ Ea+ (resp. x → a, x ∈ E ∩ Ea− ). It means that x tends to a in Ewhile remaining larger (resp.
smaller) than a.If+−E = E∞= {x ∈ R | c < x} resp. E = E∞= {x ∈ R | x < c} ,we write x → +∞ (resp. x → −∞) instead of x → ∞, x ∈ E and say that x tendsto positive infinity (resp. x tends to negative infinity).The notation E x → +∞ (resp. E x → −∞) will be used instead of+ (resp. x → ∞, x ∈ E ∩ E − ).x → ∞, x ∈ E ∩ E∞∞When E = N, we shall write (when no confusion can arise), as is customary inthe theory of limits of sequences, n → ∞ instead of x → ∞, x ∈ N.We remark that all the bases just listed have the property that the intersection oftwo elements of the base is itself an element of the base, not merely a set containingan element of the base. We shall meet with other bases in the study of functionsdefined on sets different from the real line.1111 For example, the set of open disks (not containing their boundary circles) containing a givenpoint of the plane is a base.
The intersection of two elements of the base is not always a disk, butalways contains a disk from the collection.1283LimitsWe note also that the term “base” used here is an abbreviation for what is calleda “filter base”, and the limit over a base that we introduce below is, as far as analysisis concerned, the most important part of the concept of a limit over a filter,12 createdby the modern French mathematician H. Cartan.b. The Limit of a Function over a BaseDefinition 12 Let f : X → R be a function defined on a set X and B a base in X.A number A ∈ R is called the limit of the function f over the base B if for everyneighborhood V (A) of A there is an element B ∈ B whose image f (B) is containedin V (A).If A is the limit of f : X → R over the base B, we writelim f (x) = A.BWe now repeat the definition of the limit over a base in logical symbols:lim f (x) = A := ∀V (A) ∃B ∈ B f (B) ⊂ V (A) .BSince we are considering numerical-valued functions at the moment, it is usefulto keep in mind the following form of this fundamental definition:lim f (x) = A := ∀ε > 0 ∃B ∈ B ∀x ∈ B f (x) − A < ε .BIn this form we take an ε-neighborhood (symmetric with respect to A) insteadof an arbitrary neighborhood V (A).
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