Handbook of mathematics for engineers and scienteists part 40 pot

6.1. BASIC CONCEPTS OF MAT H E MAT I C A L ANALYSIS 241 3. If there exist lim x→a f(x) and lim x→a g(x), then lim x→a  f(x) g(x)  = lim x→a f(x) lim x→a g(x); lim x→a cf(x)=c lim x→a f(x)(c = const); lim x→a f(x) ⋅ g(x) = lim x→a f(x) ⋅ lim x→a g(x); lim x→a f(x) g(x) = lim x→a f(x) lim x→a g(x)  if g(x) ≠ 0, lim x→a g(x) ≠ 0  . 4. Let f(x) ≤ g(x) in a neighborhood of a point a (x ≠ a). Then lim x→a f(x) ≤ lim x→a g(x), provided that these limits exist. 5. If f (x) ≤ g(x) ≤ h(x) in a neighborhood of a point a and lim x→a f(x) = lim x→a h(x)=b, then lim x→a g(x)=b. These properties hold also for one-sided limits. 6.1.3-3. Limits of some functions. First noteworthy limit: lim x→0 sin x x = 1. Second noteworthy limit: lim x→∞  1 + 1 x  x = e. Some other frequently used limits: lim x→0 (1 + x) n – 1 x = n, lim x→∞ a n x n + a n–1 x n–1 + ···+ a 1 x + a 0 b n x n + b n–1 x n–1 + ···+ b 1 x + b 0 = a n b n , lim x→0 1 –cosx x 2 = 1 2 , lim x→0 tan x x = 1, lim x→0 arcsin x x = 1, lim x→0 arctan x x = 1, lim x→0 e x – 1 x = 1, lim x→0 a x – 1 x =lna, lim x→0 ln(1 + x) x = 1, lim x→0 log a (1 + x) x =log a e, lim x→0 sinh x x = 1, lim x→0 tanh x x = 1, lim x→0 arcsinh x x = 1, lim x→0 arctanh x x = 1, lim x→+0 x a ln x = 0, lim x→+∞ x –a ln x = 0, lim x→+∞ x a e –x = 0, lim x→+0 x x = 1, where a > 0 and b n ≠ 0.  See Paragraph 6.2.3-2, where L’Hospital rules for calculating limits with the help of derivatives are given. 6.1.3-4. Asymptotes of the graph of a function. An asymptote of the graph of a function y = f (x) is a straight line whose distance from the point (x, y) on the graph of y = f (x) tends to zero if at least one of the coordinates (x, y) tends to zero. 242 LIMITS AND DERIVATIVES The line x = a is a vertical asymptote of the graph of the function y = f(x) if at least one of the one-sided limits of f (x)asx → a 0 is equal to +∞ or –∞. The line y = kx + b is an oblique asymptote of the graph of y = f (x) if at least one of the limit relations holds: lim x→+∞ [f(x)–kx – b]=0 or lim x→–∞ [f(x)–kx – b]=0. If there exist finite limits lim x→+∞ f(x) x = k, lim x→+∞ [f(x)–kx]=b,(6.1.3.2) then the line y = kx + b is an oblique asymptote of the graph for x → +∞ (in a similar way, one defines an asymptote for x → –∞). Example. Let us find the asymptotes of the graph of the function y = x 2 x – 1 . 1 ◦ . The graph has a vertical asymptote x = 1, since lim x→1 x 2 x – 1 = ∞. 2 ◦ . Moreover, for x → ∞, there is an oblique asymptote y = kx + b whose coefficients are determined by the formulas (6.1.3.2): k = lim x→ ∞ x x – 1 = 1, b = lim x→ ∞  x 2 x – 1 – x  = lim x→ ∞ x x – 1 = 1. Thus, the equation of the oblique asymptote has the form y = x + 1. Fig. 6.1 shows the graph of the function under consideration and its asymptotes. O x y yx= +1 1 3 2 5 4 1 2 3 4 5 6 7 y = x 1 x 2 2 1 1 2 3 Figure 6.1. The graph and asymptotes of the function y = x 2 x – 1 . 6.1.4. Infinitely Small and Infinitely Large Functions 6.1.4-1. Definitions. A function f (x) is called infinitely small for x → a if lim x→a f(x)=0. 6.1. BASIC CONCEPTS OF MAT H E MAT I C A L ANALYSIS 243 A function f (x)issaidtobeinfinitely large for x → a if for any K > 0 the inequality |f(x)| > K holds for all x ≠ a in a small neighborhood of the point a. In this case, one writes f(x) →∞as x → a or lim x→a f(x)=∞. (In these definitions, a is a finite number or any of the symbols ∞,+∞,–∞ .) If f(x)isinfinitely large for x → a and f(x)>0 (f(x)<0)in a neighborhood of a (for x ≠ a), one writes lim x→a f(x)=+∞ (resp., lim x→a f(x)=–∞). 6.1.4-2. Properties of infinitely small and infinitely large functions. 1. The sum and the product of finitely many infinitely small functions for x → a is an infinitely small function. 2. The product of an infinitely small function f(x)forx → a and a function g(x)which is bounded in a neighborhood U of the point a (i.e., |g(x)| < M for all x U,whereM > 0 is a constant) is an infinitely small function. 3. lim x→a f(x)=b if and only if f (x)=b + g(x), where g(x)isinfinitely small for x → a. 4. A function f(x)isinfinitely large at some point if and only if the function g(x)= 1 f(x) is infinitely small at the same point. 6.1.4-3. Comparison of infinitely large quantities. Symbols of the order: O and o. Functions f (x)andg(x)thatareinfinitely small for x → a are called equivalent near a if lim x→a f(x) g(x) = 1. In this case one writes f(x) ∼ g(x). Examples of equivalent infinitely small functions: (1 + ε) n – 1 ∼ nε, a ε – 1 ∼ ε ln a,log a (1 + ε) ∼ ε log a e, sin ε ∼ ε,tanε ∼ ε, 1 –cosε ∼ 1 2 ε 2 ,arcsinε ∼ ε,arctanε ∼ ε, where ε = ε(x)isinfinitely small for x → a. Functions f (x)andg(x) are said to be of the same order for x → a, and one writes f(x)=O  g(x)  if lim x→a f(x) g(x) = K, 0 < |K| < ∞.* A functionf(x)isofahigher order of smallness compared with g(x)forx → a if lim x→a f(x) g(x) = 0, and in this case, one writes f(x)=o  g(x)  . 6.1.5. Continuous Functions. Discontinuities of the First and the Second Kind 6.1.5-1. Continuous functions. A function f(x) is called continuous at a point x = a if it is defined in that point and its neighborhood and lim x→a f(x)=f(a). For continuous functions, a small variation of their argument Δx = x – a corresponds to a small variation of the function Δy = f (x)–f (a), i.e., Δy → 0 as Δx → 0. (This property is often used as a definition of continuity.) * There is another definition of the symbol O. Namely, f(x)=O  g(x)  for x → a if the inequality |f(x)| ≤ K|g(x)|, K = const, holds in some neighborhood of the point a (for x ≠ a). 244 LIMITS AND DERIVATIVES A function f (x) is called right-continuous at a point x = a if it is defined in that point (and to its right) and lim x→a+0 f(x)=f(a). A function f (x) is called left-continuous at a point x = a if it is defined in that point (and to its left) and lim x→a–0 f(x)=f(a). 6.1.5-2. Properties of continuous functions. 1. Suppose that functions f (x)andg(x) are continuous at some point a. Then the functions f (x) g(x), cf (x), f(x)g(x), f(x) g(x) (g(a) ≠ 0) are also continuous at a. 2. Suppose that a function f (x) is continuous on the segment [a, b] and takes values of different signs at its endpoints, i.e., f(a)f(b)<0. Then there is a point c between a and b at which f(x)vanishes: f(c)=0 (a < c < b). 3. If f (x) is continuous at a point a and f (a)>0 (resp., f(a)<0), then there is δ > 0 such that f(x)>0 (resp., f(x)<0)forallx (a – δ,a + δ). 4. Any function f(x) that is continuous at each point of a segment [a, b] attains its largest and its smallest values, M and m, on that segment. 5. A function f(x) that is continuous on a segment [a, b] takes any value c [m, M]on that segment, where m and M are, respectively, its smallest and its largest values on [a, b]. 6. If f(x) is continuous and increasing (resp., decreasing) on a segment [a, b], then on the segment  f(a), f(b)  (resp.,  f(b), f(a)  ) the inverse function x = g(y) exists, and is continuous and increasing (resp., decreasing). 7. If u(x) is continuous at a point a and f(u) is continuous at b =u(a), then the composite function f  u(x)  is continuous at a. Remark. Any elementary function is continuous at each point of its domain. 6.1.5-3. Points of discontinuity of a function. A point a is called a point of discontinuity of the first kind for a function f(x)ifthereexist finite one-sided limits f(a+0)andf(a–0), but the relations lim x→a+0 f(x) = lim x→a–0 f(x)=f(a) do not hold. The value |f(a + 0)–f(a – 0)| is called the jump of the function at the point a. In particular, if f(a+0)=f(a–0) ≠ f(a), then a is called a point of removable discontinuity. Examples of function with discontinuities of the first kind. 1. The function f(x)=  0 for x < 0, 1 for x ≥ 0 has a jump equal to 1 at the discontinuity point x = 0. 2. The function f(x)=  0 for x ≠ 0, 1 for x = 0 has a removable discontinuity at the point x = 0. A point a is called a point of discontinuity of the second kind if at least one of the one-sided limits f(a + 0)orf(a – 0) does not exist or is equal to infinity. Examples of functions with discontinuities of the second kind. 1. The function f(x)=sin 1 x has a second-kind discontinuity at the point x = 0 (since this function has no one-sided limits as x → 0). 2. The function f(x)=1/x has an infinite jump at the point x = 0. 6.1. BASIC CONCEPTS OF MAT H E MAT I C A L ANALYSIS 245 6.1.5-4. Properties of monotone functions at points of discontinuity. Any monotone function f (x) always has a left-hand limit and a right-hand limit at its discontinuity point x = x 0 ; moreover, if f(x) is a nonincreasing function, then f(x 0 – 0) ≥ f (x 0 ) ≥ f (x 0 + 0); if f(x) is a nondecreasing function, then f(x 0 – 0) ≤ f(x 0 ) ≤ f (x 0 + 0). 6.1.6. Convex and Concave Functions 6.1.6-1. Definition of convex and concave functions. 1 ◦ . A function f(x)defined and continuous on a segment [a, b] is called convex (or convex downward)ifforanyx 1 , x 2 in [a, b], the Jensen inequality holds: f  x 1 + x 2 2  ≤ f(x 1 )+f (x 2 ) 2 .(6.1.6.1) The geometrical meaning of convexity is that all points of the graph curve between two graph points lie below or on the rectilinear segment joining the two graph points (see Fig. 6.2 a). OO 2211 xx yy xxxx yfx= () yfx= () ()a ()b 1 fx() 2 fx() 1 fx() f () 2 21 x+x 2 21 fx()+fx() 2 fx() f () 2 21 x+x 2 21 fx()+fx() 2 21 x+x 2 21 x+x Figure 6.2. Graphs of convex (a) and concave (b) functions. If for x 1 ≠ x 2 , condition (6.1.6.1) holds with < instead of ≤, then the function f(x)is called strictly convex. 2 ◦ . A function f(x)defined and continuous on a segment [a, b] is called concave (or convex upward)ifforanyx 1 , x 2 in [a, b] the following inequality holds: f  x 1 + x 2 2  ≥ f(x 1 )+f (x 2 ) 2 .(6.1.6.2) The geometrical meaning of concavity is that all points of the graph curve between two graph points lie above or on the rectilinear segment joining the two graph points (see Fig. 6.2 b). If for x 1 ≠ x 2 , condition (6.1.6.2) holds with > instead of ≥, then the function f(x)is called strictly concave. 246 LIMITS AND DERIVATIVES 6.1.6-2. Generalized Jensen inequalities. The inequalities (6.1.6.1) and (6.1.6.2) admit the following generalizations: f(q 1 x 1 + ···+ q n x n ) ≤ q 1 f(x 1 )+···+ q n x n for a convex function, f(q 1 x 1 + ···+ q n x n ) ≥ q 1 f(x 1 )+···+ q n x n for a concave function, where q 1 , , q n are arbitrary positive numbers such that q 1 + ···+ q n = 1,andx 1 , , x n are arbitrary points of the segment [a, b]. 6.1.6-3. Properties of convex and concave functions. 1. The product of a convex (concave) function and a positive constant is a convex (concave) function. 2. The sum of two or more convex (concave) functions is a convex (concave) function. 3. If ϕ(u) is a convex increasing function and u = f (x) is a convex function, then the composite function ϕ(f (x)) is convex. Some other properties of composite functions: ϕ(u) is convex and decreasing, u = f (x) is concave =⇒ ϕ(f(x)) is convex, ϕ(u) is concave and increasing, u = f(x) is concave =⇒ ϕ(f(x)) is concave, ϕ(u) is concave and decreasing, u = f (x) is convex =⇒ ϕ(f(x)) is concave. 4. A non-constant convex (resp., concave) function f (x)onasegment[a, b] cannot attain its largest (resp., smallest) value inside the segment. 5. If y = f(x)andx = g(y) are single-valued mutually inverse functions (on the corre- sponding intervals), then the following properties hold: f(x) is convex and increasing ⇐⇒ g(y) is concave and increasing, f(x) is convex and decreasing ⇐⇒ g(y) is convex and decreasing, f(x) is concave and increasing ⇐⇒ g(y) is convex and increasing, f(x) is concave and decreasing ⇐⇒ g(y) is concave and decreasing. 6. A function f (x) that is continuous on a segment [a, b] and twice differentiable on the interval (a, b) is convex downward (resp., convex upward) if and only if f  (x) ≥ 0 (resp., f  (x) ≤ 0) on that interval. 7. Any convex function f(x) satisfying the condition f (x 0 )=0 can be represented as the integral f(x)=  x x 0 h(t) dt, where h(t) is a nondecreasing right-continuous function. 6.1.7. Functions of Bounded Variation 6.1.7-1. Definition of a function of bounded variation. 1 ◦ .Letf(x) be a function definedonafinite segment [a, b]. Consider an arbitrary partition of the segment by the points a = x 0 < x 1 < x 2 < ···< x n–1 < x n = b 6.1. BASIC CONCEPTS OF MAT H E MAT I C A L ANALYSIS 247 and construct the sum v = n–1  k=0   f(x k+1 )–f(x k )   (6.1.7.1) whose terms are absolute values of the increments of f(x) on each segment of the partition. If, for all partitions, the sums (6.1.7.1) are bounded by a constant independent of the partition, one says that the function f(x)hasbounded variation on the segment [a, b]. The supremum of all such sums over all partitions is called the total variation of the function f(x) on the segment [a, b]. The total variation is denoted by b V a f(x)=sup{v}. A function f (x) is said to have bounded variation on the infinite interval [a, ∞)ifitis a function of bounded variation on any fi nite segment [a, b] and its total variation on [a, b] is bounded by a constant independent of b.Bydefinition, ∞ V a f(x)=sup b>a  b V a f(x)  . 2 ◦ . In the above definitions, the continuity of the function f(x) is not mentioned. A contin- uous function (without additional conditions) may have bounded or unbounded variation. Example. Consider the continuous function f(x)=  x cos π 2x if x ≠ 0, 0 if x = 0 and the partition of the segment [0, 1] by the points 0 < 1 2n < 1 2n – 1 < ···< 1 3 < 1 2 < 1. Then the sums (6.1.7.1) corresponding to this partition have the form v n = 1 + 1 2 + ···+ 1 n →∞ as n →∞. Therefore, 1 V 0 f(x)=∞. 6.1.7-2. Classes of functions of bounded variation. Next, we list some common classes of functions of bounded variation. 1. Any bounded monotone function has bounded variation. Its total variation on the segment [a, b]isdefined by b V a f(x)=|f(b)–f(a)|. Remark. The last statement is true for infinite intervals (–∞, a]and[a, ∞); in the latter case, the total variation is equal to ∞ V a f(x)=|f (∞)–f(a)|. 2. Suppose that f (x) is a bounded function on [a, b] and this segment can be divided into finitely many parts [a k , a k+1 ](k = 0, 1, , m – 1; a 0 = a, a m = b), so that the function f(x) is monotone on each part. Then f(x) has bounded variation on [a, b]. Remark. This statement is also true for infinite segments. . small for x → a. Functions f (x)andg(x) are said to be of the same order for x → a, and one writes f(x)=O  g(x)  if lim x→a f(x) g(x) = K, 0 < |K| < ∞.* A functionf(x)isofahigher order of. point of its domain. 6.1.5-3. Points of discontinuity of a function. A point a is called a point of discontinuity of the first kind for a function f(x)ifthereexist finite one-sided limits f(a+0)andf(a–0),. Properties of infinitely small and infinitely large functions. 1. The sum and the product of finitely many infinitely small functions for x → a is an infinitely small function. 2. The product of an infinitely