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Since the extreme terms here tend to e as x → +∞, it follows fromTheorem 3 on the properties of a limit that limx→+∞ (1 + x1 )x = e.Using Theorem 5 on the limit of a composite function, we now show thatlimx→−∞ (1 + x1 )x = e.1343LimitsProof We writelimx→−∞11+xx1 (−t)1 −t= lim= lim 1 −=1+t→+∞(−t)→−∞(−t)tt1== lim 1 +t→+∞t −1t−111== lim 1 +lim 1 +t→+∞t→+∞t −1t −1t−111 u= lim 1 += lim 1 += e.t→+∞u→+∞t −1uWhen we take account of the substitutions u = t − 1 and t = −x, these equalitiescan be justified in reverse order (!) using Theorem 5. Indeed, only after we havearrived at the limit limu→+∞ (1 + u1 )u , whose existence has already been proved,does the theorem allow us to assert that the preceding limit also exists and has thesame value.

Then the limit before that one also exists, and by a finite number of suchtransitions we finally arrive at the original limit. This is a very typical example of theprocedure for using the theorem on the limit of a composite function in computinglimits.Thus, we have1 x1 xlim 1 += e = lim 1 +.x→−∞x→+∞xxIt follows that limx→∞ (1 + x1 )x = e. Indeed, let ε > 0 be given.Since limx→−∞ (1 + x1 )x = e, there exists c1 ∈ R such that |(1 + x1 )x − e| < ε forx < c1 .Since limx→+∞ (1 + x1 )x = e, there exists c2 ∈ R such that |(1 + x1 )x − e| < ε forc2 < x.Then for |x| > c = max{|c1 |, |c2 |} we have |(1 + x1 )x − e| < ε, which verifies thatlimx→∞ (1 + x1 )x = e.Example 21 We shall show thatlim (1 + t)1/t = e.t→0Proof After the substitution x = 1/t, we return to the limit considered in the preceding example.Example 22limx→+∞x= 0,qxif q > 1.3.2 The Limit of a Function135Proof We know (see Example 11 in Sect. 3.1) that limn→∞ qnn = 0 if q > 1.Now, as in Example 20, we can consider the auxiliary mapping f : R+ → Ngiven by the function [x] (the integer part of x).

Using the inequalities1 [x]x[x] + 1·<< [x]+1 · qq q [x] q xqand taking account of the theorem on the limit of a composite function, we find thatthe extreme terms here tend to 0 as x → +∞. We conclude that limx→+∞ qxx = 0. Example 23loga x= 0.x→+∞xlimProof Let a > 1. Set t = loga x, so that x = a t . From the properties of the exponential function and the logarithm (taking account of the unboundedness of a n forn ∈ N) we have (x → +∞) ⇔ (t → +∞).

Using the theorem on the limit of acomposite function and the result of Example 22, we obtainloga xt= lim t = 0.x→+∞t→+∞ axlimIf 0 < a < 1 we set −t = loga x, x = a −t . Then (x → +∞) ⇔ (t → +∞), andsince 1/a > 1, we again havelimx→+∞loga x−tt= lim −t = − lim= 0.t→+∞ at→+∞ (1/a)txc. The Limit of a Monotonic FunctionWe now consider a special class of numerical-valued functions, but one that is veryuseful, namely the monotonic functions.Definition 17 A function f : E → R defined on a set E ⊂ R is said to beincreasing on E if∀x1 , x2 ∈ E x1 < x2 ⇒ f (x1 ) < f (x2 ) ;nondecreasing on E if∀x1 , x2 ∈ E x1 < x2 ⇒ f (x1 ) ≤ f (x2 ) ;nonincreasing on E if∀x1 , x2 ∈ E x1 < x2 ⇒ f (x1 ) ≥ f (x2 ) ;1363Limitsdecreasing on E if∀x1 , x2 ∈ E x1 < x2 ⇒ f (x1 ) > f (x2 ) .Functions of the types just listed are said to be monotonic on the set E.Assume that the numbers (or symbols −∞ or +∞) i = inf E and s = sup E arelimit points of the set E, and let f : E → R be a monotonic function on E.

Thenthe following theorem holds.Theorem 6 (Criterion for the existence of a limit of a monotonic function) A necessary and sufficient condition for a function f : E → R that is nondecreasing on theset E to have a limit as x → s, x ∈ E, is that it be bounded above. For this functionto have a limit as x → i, x ∈ E, it is necessary and sufficient that it be boundedbelow.Proof We shall prove this theorem for the limit limEx→s f (x).If this limit exists, then, like any function having a limit, the function f is ultimately bounded over the base E x → s.Since f is nondecreasing on E, it follows that f is bounded above. In fact, wecan even assert that f (x) ≤ limEx→s f (x).

That will be clear from what follows.Let us pass to the proof of the existence of the limit limEx→s f (x) when f isbounded above.Given that f is bounded above, we see that there is a least upper bound of thevalues that the function assumes on E \ {s}. Let A = supx∈E\{s} f (x). We shallshow that limEx→s f (x) = A. Given ε > 0, we use the definition of the least upperbound to find a point x0 ∈ E \ {s} for which A − ε < f (x0 ) ≤ A. Then, since fis nondecreasing on E, we have A − ε < f (x) ≤ A for x0 < x ∈ E \ {s}. But theset {x ∈ E | x0 < x < s} is obviously an element of the base x → s, x ∈ E (sinces = sup E). Thus we have proved that limEx→s f (x) = A.For the limit limEx→i f (x) the reasoning is analogous.

In this case we havelimEx→i f (x) = infx∈E\{i} f (x).d. Comparison of the Asymptotic Behavior of FunctionsWe begin this discussion with some examples to clarify the subject.Let π(x) be the number of primes not larger than a given number x ∈ R. Although for any fixed x we can find (if only by explicit enumeration) the value ofπ(x), we are nevertheless not in a position to say, for example, how the functionπ(x) behaves as x → +∞, or, what is the same, what the asymptotic law of distribution of prime numbers is. We have known since the time of Euclid that π(x) → +∞as x → +∞, but the proof that π(x) grows approximately like lnxx was achievedonly in the nineteenth century by P.L. Chebyshev.1313 P.L. Chebyshev (1821–1894) – outstanding Russian mathematician and specialist in theoreticalmechanics, the founder of a large mathematical school in Russia.3.2 The Limit of a Function137When it becomes necessary to describe the behavior of a function near somepoint (or near infinity) at which, as a rule, the function itself is not defined, we saythat we are interested in the asymptotics or asymptotic behavior of the function in aneighborhood of the point.The asymptotic behavior of a function is usually characterized using a secondfunction that is simpler or better studied and which reproduces the values of thefunction being studied in a neighborhood of the point in question with small relativeerror.Thus, as x → +∞, the function π(x) behaves like lnxx ; as x → 0, the functionsin xx behaves like the constant function 1.

When we speak of the behavior of thefunction x 2 + x + sin x1 as x → ∞, we shall obviously say that it behaves basicallylike x 2 , while in speaking of its behavior as x → 0, we shall say it behaves likesin x1 .We now give precise definitions of some elementary concepts involving theasymptotic behavior of functions. We shall make systematic use of these conceptsat the very first stage of our study of analysis.Definition 18 We shall say that a certain property of functions or a certain relationbetween functions holds ultimately over a given base B if there exists B ∈ B onwhich it holds.We have already interpreted the notion of a function that is ultimately constantor ultimately bounded in a given base in this sense.

In the same sense we shall sayfrom now on that the relation f (x) = g(x)h(x) holds ultimately between functionsf , g, and h. These functions may have at the outset different domains of definition,but if we are interested in their asymptotic behavior over the base B, all that mattersto us is that they are all defined on some element of B.Definition 19 The function f is said to be infinitesimal compared with the functiong over the base B, and we write f =B o(g) or f = o(g) over B if the relation f (x) =α(x)g(x) holds ultimately over the B, where α(x) is a function that is infinitesimalover B.Example 24 x 2 = o(x) as x → 0, since x 2 = x · x.Example 25 x = o(x 2 ) as x → ∞, since ultimately (as long as x = 0), x =1x· x2.From these examples one must conclude that it is absolutely necessary to indicatethe base over which f = o(g).The notation f = o(g) is read “f is little-oh of g”.It follows from the definition, in particular, that the notation f =B o(1), whichresults when g(x) ≡ 1, means simply that f is infinitesimal over B.Definition 20 If f =B o(g) and g is itself infinitesimal over B, we say that f is aninfinitesimal of higher order than g over B.1383Example 26 x −2 =1x2is an infinitesimal of higher order than x −1 =1xLimitsas x → ∞.Definition 21 A function that tends to infinity over a given base is said to be aninfinite function or simply an infinity over the given base.Definition 22 If f and g are infinite functions over B and f =B o(g), we say thatg is a higher order infinity than f over B.Example 27 x1 → ∞ as x → 0, x12 → ∞ as x → 0 and x1 = o( x12 ).

Thereforea higher order infinity than x1 as x → 0.At the same time, as x → ∞, x 2 is a higher order infinity than x.1x2isIt should not be thought that we can characterize the order of every infinity orinfinitesimal by choosing some power x n and saying that it is of order n.Example 28 We shall show that for a > 1 and any n ∈ Zxn= 0,x→+∞ a xlimthat is, x n = o(a x ) as x → +∞.Proof If n ≤0 the assertion is obvious. If n ∈ N, then, setting q =nq > 1 and xa x = ( qxx )n , and therefore√na, we have nxxnxx=lim= lim x · . . . · lim x = 0.xxx→+∞ ax→+∞ qx→+∞ qx→+∞ qlimn factorsWe have used (with induction) the theorem on the limit of a product and the resultof Example 22.Thus, for any n ∈ Z we obtain x n = o(a x ) as x → +∞ if a > 1.Example 29 Extending the preceding example, let us show thatxα=0x→+∞ a xlimfor a > 1 and any α ∈ R, that is, x α = o(a x ) as x → +∞.Proof Indeed, let us choose n ∈ N such that n > α.

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