DOCSLIB.ORG

Path Spaces, Continuous Tensor Products, and E0

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Path Spaces, Continuous Tensor Products, and E0 PATH SPACES, CONTINUOUS TENSOR PRODUCTS, AND E0-SEMIGROUPS William Arveson Department of Mathematics University of California Berkeley CA 94720, USA Abstract. We classify all continuous tensor product systems of Hilbert spaces which are “infinitely divisible” in the sense that they have an associated logarithmic struc- ture. These results are applied to the theory of E0-semigroups to deduce that every E0-semigroup possessing sufficiently many “decomposable” operators must be cocy- cle conjugate to a CCR flow. A path space is an abstraction of the set of paths in a topological space, on which there is given an associative rule of concatenation. A metric path space is a pair 2 (P, g) consisting of a path space P and a function g : P → C which behaves as if it were the logarithm of a multiplicative inner product. The logarithmic structures associated with infinitely divisible product systems are such objects. The preceding results are based on a classification of metric path spaces. Contents Introduction Part I. Path Spaces 1. Definitions and examples 2. Additive forms and multiplicative forms 3. Exactness of cocycles 4. Classification of additive forms Part II. Continuous Tensor Products 5. Continuity of the modulus 6. Inner products of decomposable vectors 7. Continuity and normalization 8. Continuous logarithms 9. Infinite divisibility 10. Existence of measurable propagators Part III. Applications 11. Decomposable continuous tensor products 12. Decomposable E0-semigroups 1991 Mathematics Subject Classification. Primary 46L40; Secondary 81E05. Key words and phrases. von Neumann algebras, semigroups, automorphism groups. This research was supported in part by NSF grant DMS92-43893 Typeset by AMS-TEX 1 2 WILLIAM ARVESON Introduction. Let α = {αt : t > 0} be an E0-semigroup. That is, each αt is a normal unit-preserving ∗-endomorphism of B(H) such that αs ◦ αt = αs+t, and which is continuous in the sense that for all A ∈ B(H), ξ, η ∈ H, hαt(A)ξ, ηi is continuous in t. A unit for α is a strongly continuous semigroup U = {U(t): t ≥ 0} of bounded operators on H satisfying αt(A)U(t) = U(t)A, for every t ≥ 0, A ∈ B(H). α is called completely spatial if there is a t > 0 such that H is spanned by the ranges of all operator products of the form U1(t1)U2(t2) ...Un(tn) where U1,U2,...,Un are units, t1, t2, . , tn are nonnegative reals summing to t, and n is an arbitrary positive integer. Completely spatial E0-semigroups are those which have “sufficiently many” units. In [2, section 7], completely spatial E0-semigroups are shown to be completely classified up to cocycle conjugacy by their numerical index. This is established at the level of continuous tensor product systems. In somewhat more detail, a product system is a measurable family of separable Hilbert spaces E = {Et : t > 0} which is endowed with an associative multiplication that “acts like tensoring” in the sense 0 0 0 0 that for any choices x, x ∈ Es, y, y ∈ Et, the products xy, x y both belong to Es+t and we have hxy, x0y0i = hx, x0i hy, y0i , and Es+t = spanEsEt. In addition, there is a third key axiom which plays the role of local triviality for Hermitian vector bundles. The precise definition of product systems can be found in [2, Definition 1.4]. The intuition is that Et resembles a continuous tensor product O (I.1) Et = Hs,Hs = H 0<s<t of copies of a single Hilbert space H. However, this heuristic picture of continuous tensor products cannot be pushed too far. While the formula (I.1) can be made precise for certain standard examples, it is also known that there are many product systems for which the “germ” H fails to exist. th Every E0-semigroup α determines a product system Eα. The t Hilbert space Eα(t) is the linear space of operators Eα(t) = {T ∈ B(H): αt(A)T = T A, ∀A ∈ B(H)}, with inner product defined by ∗ hS, T i 1 = T S, S, T ∈ Eα(t), and with multiplication given by ordinary operator multiplication. It is known that two E0-semigroups are cocycle conjugate iff their product systems Eα, Eβ PATH SPACES, CONTINUOUS TENSOR PRODUCTS, AND E0-SEMIGROUPS 3 are isomorphic [2, Corollary of Theorem 3.18]; moreover, every abstract product system is associated with some E0-semigroup [5, Corollary 5.17]. These remarks show that, up to cocycle conjugacy, the theory of E0-semigroups is equivalent to the theory of continuous tensor product systems. Consequently, a central component of our approach to E0-semigroups has been to develop the theory of product systems. The classification of E0-semigroups described above was obtained by classifying product systems which possess a property that corresponds to complete spatiality of their product systems (this was called divisibility in [2]). In this paper, we extend that result so as to include certain product systems which do not appear to contain any units a priori, but which do contain sufficiently many vectors that resemble “elementary tensors”. Such product systems are called decomposable. The corresponding property of E0-semigroups is described as follows. Fix t > 0. An operator T ∈ Eα(t) is called decomposable if, for every 0 < s < t there are operators A ∈ Eα(s), B ∈ Eα(t − s) such that T = AB. Let D(t) denote the set of all decomposable operators in Eα(t). It is easy to see that if H is spanned by vectors of the form D(t0)H = {T ξ : T ∈ D(t0), ξ ∈ H} for some particular t0 > 0, then we have (I.2) H = [D(t)H] for every t > 0. α is called decomposable if (I.2) is valid for some t > 0. Notice that if U1,U2,...,Un are units for α and t1, t2, . , tn are positive numbers summing to t, then every operator product of the form U1(t1)U2(t2) ...Un(tn) is a decomposable operator in Eα(t) because each Uj is a semigroup. Thus, every completely spatial E0-semigroup is decomposable. On the other hand, decomposable E0-semigroups are not required to contain any units a priori. The results of this paper imply that decomposable E0-semigroups are completely spatial, and are therefore classified to cocycle conjugacy by their nu- merical index. In particular, decomposable E0-semigroups necessarily have plenty of units. Decomposability translates into an important property of abstract product sys- tems, and we want to discuss this property and its role in structural issues. Let E = {E(t): t > 0} be an abstract product system and fix t > 0. A nonzero vector x ∈ E(t) is called decomposable if for every s ∈ (0, t) there are vectors a ∈ E(s), b ∈ E(t − s) such that x = ab. Let D(t) be the set of all decomposable vectors in E(t). D(t) can be empty. But if it is large enough that it spans E(t) for some particular t, then it is easy to show that (I.3) E(t) = spanD(t) for every positive t. A product system satisfying (I.3) for every positive t is called decomposable. It is also easy to show that if E is isomorphic to the product system Eα of some E0-semigroup α, then E is decomposable iff α is (Proposition 12.2). 4 WILLIAM ARVESON If we think of the multiplication in a product system E as representing the tensor product operation, then the vectors in D(t) are “elementary tensors”. In the heuristic picture in which one thinks of E(t) as a continuous tensor product O E(t) = Hs,Hs = H 0<s<t of copies of a Hilbert space H, the decomposable vectors have a corresponding heuristic representation (I.4) x = ⊗ xs 0<s<t where xs is a nonzero vector in H for every s ∈ (0, t). Since finite tensor products of Hilbert spaces of the form H ⊗ H ⊗ · · · ⊗ H | {z } n times are always spanned by vectors of the form x = x1 ⊗ x2 ⊗ · · · ⊗ xn with xi ∈ H, decomposability appears as a natural property of product systems. However, notice that Theorem 11.1 below implies that only the simplest product systems are decomposable. Product systems occur naturally in probability theory. For example, any ran- dom distribution over the interval (0, ∞) having stationary independent increments gives rise to a product system (see [2, pp 14–16] for an example based on Poisson processes). Some of these examples (such those arising from Poisson processes) do not appear to have “enough” units to be divisible in the sense of [2]. Nevertheless, they do. They do because in constructing these examples, one essentially constructs the sets D(t), t > 0, and then defines the product system E by setting E(t) = spanD(t), t > 0. Such a product system is clearly decomposable and thus, by Theorem 11.1 below, is simply one of the standard ones. Our approach to decomposable product systems is based on the concept of a metric path space. Roughly speaking, a metric path space is a structure which behaves as if it were the “logarithm” of a product system. Path spaces are very general objects, and we give a variety of examples. The most natural constructions of product systems all begin with a metric path space of some kind...an example of this being the construction starting with a Poisson process as above. We give an essentially complete classification of metric path spaces in part I. In part II, we show that every decomposable product system is associated with an essentially unique metric path space, and in part III these results are applied to the classification of product systems and E0-semigroups.
Load more

Recommended publications
  • Section 8.8: Improper Integrals
    Section 8.8: Improper Integrals One of the main applications of integrals is to compute the areas under curves, as you know. A geometric question. But there are some geometric questions which we do not yet know how to do by calculus, even though they appear to have the same form. Consider the curve y = 1=x2. We can ask, what is the area of the region under the curve and right of the line x = 1? We have no reason to believe this area is finite, but let's ask. Now no integral will compute this{we have to integrate over a bounded interval. Nonetheless, we don't want to throw up our hands. So note that b 2 b Z (1=x )dx = ( 1=x) 1 = 1 1=b: 1 − j − In other words, as b gets larger and larger, the area under the curve and above [1; b] gets larger and larger; but note that it gets closer and closer to 1. Thus, our intuition tells us that the area of the region we're interested in is exactly 1. More formally: lim 1 1=b = 1: b − !1 We can rewrite that as b 2 lim Z (1=x )dx: b !1 1 Indeed, in general, if we want to compute the area under y = f(x) and right of the line x = a, we are computing b lim Z f(x)dx: b !1 a ASK: Does this limit always exist? Give some situations where it does not exist. They'll give something that blows up.
    [Show full text]
  • Notes on Calculus II Integral Calculus Miguel A. Lerma
    Notes on Calculus II Integral Calculus Miguel A. Lerma November 22, 2002 Contents Introduction 5 Chapter 1. Integrals 6 1.1. Areas and Distances. The Definite Integral 6 1.2. The Evaluation Theorem 11 1.3. The Fundamental Theorem of Calculus 14 1.4. The Substitution Rule 16 1.5. Integration by Parts 21 1.6. Trigonometric Integrals and Trigonometric Substitutions 26 1.7. Partial Fractions 32 1.8. Integration using Tables and CAS 39 1.9. Numerical Integration 41 1.10. Improper Integrals 46 Chapter 2. Applications of Integration 50 2.1. More about Areas 50 2.2. Volumes 52 2.3. Arc Length, Parametric Curves 57 2.4. Average Value of a Function (Mean Value Theorem) 61 2.5. Applications to Physics and Engineering 63 2.6. Probability 69 Chapter 3. Differential Equations 74 3.1. Differential Equations and Separable Equations 74 3.2. Directional Fields and Euler’s Method 78 3.3. Exponential Growth and Decay 80 Chapter 4. Infinite Sequences and Series 83 4.1. Sequences 83 4.2. Series 88 4.3. The Integral and Comparison Tests 92 4.4. Other Convergence Tests 96 4.5. Power Series 98 4.6. Representation of Functions as Power Series 100 4.7. Taylor and MacLaurin Series 103 3 CONTENTS 4 4.8. Applications of Taylor Polynomials 109 Appendix A. Hyperbolic Functions 113 A.1. Hyperbolic Functions 113 Appendix B. Various Formulas 118 B.1. Summation Formulas 118 Appendix C. Table of Integrals 119 Introduction These notes are intended to be a summary of the main ideas in course MATH 214-2: Integral Calculus.
    [Show full text]
  • Two Fundamental Theorems About the Definite Integral
    Two Fundamental Theorems about the Definite Integral These lecture notes develop the theorem Stewart calls The Fundamental Theorem of Calculus in section 5.3. The approach I use is slightly different than that used by Stewart, but is based on the same fundamental ideas. 1 The definite integral Recall that the expression b f(x) dx ∫a is called the definite integral of f(x) over the interval [a,b] and stands for the area underneath the curve y = f(x) over the interval [a,b] (with the understanding that areas above the x-axis are considered positive and the areas beneath the axis are considered negative). In today's lecture I am going to prove an important connection between the definite integral and the derivative and use that connection to compute the definite integral. The result that I am eventually going to prove sits at the end of a chain of earlier definitions and intermediate results. 2 Some important facts about continuous functions The first intermediate result we are going to have to prove along the way depends on some definitions and theorems concerning continuous functions. Here are those definitions and theorems. The definition of continuity A function f(x) is continuous at a point x = a if the following hold 1. f(a) exists 2. lim f(x) exists xœa 3. lim f(x) = f(a) xœa 1 A function f(x) is continuous in an interval [a,b] if it is continuous at every point in that interval. The extreme value theorem Let f(x) be a continuous function in an interval [a,b].
    [Show full text]
  • Lecture 13 Gradient Methods for Constrained Optimization
    Lecture 13 Gradient Methods for Constrained Optimization October 16, 2008 Lecture 13 Outline • Gradient Projection Algorithm • Convergence Rate Convex Optimization 1 Lecture 13 Constrained Minimization minimize f(x) subject x ∈ X • Assumption 1: • The function f is convex and continuously differentiable over Rn • The set X is closed and convex ∗ • The optimal value f = infx∈Rn f(x) is finite • Gradient projection algorithm xk+1 = PX[xk − αk∇f(xk)] starting with x0 ∈ X. Convex Optimization 2 Lecture 13 Bounded Gradients Theorem 1 Let Assumption 1 hold, and suppose that the gradients are uniformly bounded over the set X. Then, the projection gradient method generates the sequence {xk} ⊂ X such that • When the constant stepsize αk ≡ α is used, we have 2 ∗ αL lim inf f(xk) ≤ f + k→∞ 2 P • When diminishing stepsize is used with k αk = +∞, we have ∗ lim inf f(xk) = f . k→∞ Proof: We use projection properties and the line of analysis similar to that of unconstrained method. HWK 6 Convex Optimization 3 Lecture 13 Lipschitz Gradients • Lipschitz Gradient Lemma For a differentiable convex function f with Lipschitz gradients, we have for all x, y ∈ Rn, 1 k∇f(x) − ∇f(y)k2 ≤ (∇f(x) − ∇f(y))T (x − y), L where L is a Lipschitz constant. • Theorem 2 Let Assumption 1 hold, and assume that the gradients of f are Lipschitz continuous over X. Suppose that the optimal solution ∗ set X is not empty. Then, for a constant stepsize αk ≡ α with 0 2 < α < L converges to an optimal point, i.e., ∗ ∗ ∗ lim kxk − x k = 0 for some x ∈ X .
    [Show full text]
  • Calculus Terminology
    AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential
    [Show full text]
  • The Infinite and Contradiction: a History of Mathematical Physics By
    The infinite and contradiction: A history of mathematical physics by dialectical approach Ichiro Ueki January 18, 2021 Abstract The following hypothesis is proposed: \In mathematics, the contradiction involved in the de- velopment of human knowledge is included in the form of the infinite.” To prove this hypothesis, the author tries to find what sorts of the infinite in mathematics were used to represent the con- tradictions involved in some revolutions in mathematical physics, and concludes \the contradiction involved in mathematical description of motion was represented with the infinite within recursive (computable) set level by early Newtonian mechanics; and then the contradiction to describe discon- tinuous phenomena with continuous functions and contradictions about \ether" were represented with the infinite higher than the recursive set level, namely of arithmetical set level in second or- der arithmetic (ordinary mathematics), by mechanics of continuous bodies and field theory; and subsequently the contradiction appeared in macroscopic physics applied to microscopic phenomena were represented with the further higher infinite in third or higher order arithmetic (set-theoretic mathematics), by quantum mechanics". 1 Introduction Contradictions found in set theory from the end of the 19th century to the beginning of the 20th, gave a shock called \a crisis of mathematics" to the world of mathematicians. One of the contradictions was reported by B. Russel: \Let w be the class [set]1 of all classes which are not members of themselves. Then whatever class x may be, 'x is a w' is equivalent to 'x is not an x'. Hence, giving to x the value w, 'w is a w' is equivalent to 'w is not a w'."[52] Russel described the crisis in 1959: I was led to this contradiction by Cantor's proof that there is no greatest cardinal number.
    [Show full text]
  • Convergence Rates for Deterministic and Stochastic Subgradient
    Convergence Rates for Deterministic and Stochastic Subgradient Methods Without Lipschitz Continuity Benjamin Grimmer∗ Abstract We extend the classic convergence rate theory for subgradient methods to apply to non-Lipschitz functions. For the deterministic projected subgradient method, we present a global O(1/√T ) convergence rate for any convex function which is locally Lipschitz around its minimizers. This approach is based on Shor’s classic subgradient analysis and implies generalizations of the standard convergence rates for gradient descent on functions with Lipschitz or H¨older continuous gradients. Further, we show a O(1/√T ) convergence rate for the stochastic projected subgradient method on convex functions with at most quadratic growth, which improves to O(1/T ) under either strong convexity or a weaker quadratic lower bound condition. 1 Introduction We consider the nonsmooth, convex optimization problem given by min f(x) x∈Q for some lower semicontinuous convex function f : Rd R and closed convex feasible → ∪{∞} region Q. We assume Q lies in the domain of f and that this problem has a nonempty set of minimizers X∗ (with minimum value denoted by f ∗). Further, we assume orthogonal projection onto Q is computationally tractable (which we denote by PQ( )). arXiv:1712.04104v3 [math.OC] 26 Feb 2018 Since f may be nondifferentiable, we weaken the notion of gradients to· subgradients. The set of all subgradients at some x Q (referred to as the subdifferential) is denoted by ∈ ∂f(x)= g Rd y Rd f(y) f(x)+ gT (y x) . { ∈ | ∀ ∈ ≥ − } We consider solving this problem via a (potentially stochastic) projected subgradient method.
    [Show full text]
  • Mean Value, Taylor, and All That
    Mean Value, Taylor, and all that Ambar N. Sengupta Louisiana State University November 2009 Careful: Not proofread! Derivative Recall the definition of the derivative of a function f at a point p: f (w) − f (p) f 0(p) = lim (1) w!p w − p Derivative Thus, to say that f 0(p) = 3 means that if we take any neighborhood U of 3, say the interval (1; 5), then the ratio f (w) − f (p) w − p falls inside U when w is close enough to p, i.e. in some neighborhood of p. (Of course, we can’t let w be equal to p, because of the w − p in the denominator.) In particular, f (w) − f (p) > 0 if w is close enough to p, but 6= p. w − p Derivative So if f 0(p) = 3 then the ratio f (w) − f (p) w − p lies in (1; 5) when w is close enough to p, i.e. in some neighborhood of p, but not equal to p. Derivative So if f 0(p) = 3 then the ratio f (w) − f (p) w − p lies in (1; 5) when w is close enough to p, i.e. in some neighborhood of p, but not equal to p. In particular, f (w) − f (p) > 0 if w is close enough to p, but 6= p. w − p • when w > p, but near p, the value f (w) is > f (p). • when w < p, but near p, the value f (w) is < f (p). Derivative From f 0(p) = 3 we found that f (w) − f (p) > 0 if w is close enough to p, but 6= p.
    [Show full text]
  • MATH 162: Calculus II Differentiation
    MATH 162: Calculus II Framework for Mon., Jan. 29 Review of Differentiation and Integration Differentiation Definition of derivative f 0(x): f(x + h) − f(x) f(y) − f(x) lim or lim . h→0 h y→x y − x Differentiation rules: 1. Sum/Difference rule: If f, g are differentiable at x0, then 0 0 0 (f ± g) (x0) = f (x0) ± g (x0). 2. Product rule: If f, g are differentiable at x0, then 0 0 0 (fg) (x0) = f (x0)g(x0) + f(x0)g (x0). 3. Quotient rule: If f, g are differentiable at x0, and g(x0) 6= 0, then 0 0 0 f f (x0)g(x0) − f(x0)g (x0) (x0) = 2 . g [g(x0)] 4. Chain rule: If g is differentiable at x0, and f is differentiable at g(x0), then 0 0 0 (f ◦ g) (x0) = f (g(x0))g (x0). This rule may also be expressed as dy dy du = . dx x=x0 du u=u(x0) dx x=x0 Implicit differentiation is a consequence of the chain rule. For instance, if y is really dependent upon x (i.e., y = y(x)), and if u = y3, then d du du dy d (y3) = = = (y3)y0(x) = 3y2y0. dx dx dy dx dy Practice: Find d x d √ d , (x2 y), and [y cos(xy)]. dx y dx dx MATH 162—Framework for Mon., Jan. 29 Review of Differentiation and Integration Integration The definite integral • the area problem • Riemann sums • definition Fundamental Theorem of Calculus: R x I: Suppose f is continuous on [a, b].
    [Show full text]
  • Generalizing the Mean Value Theorem -Taylor's Theorem
    Generalizing the Mean Value Theorem { Taylor's theorem We explore generalizations of the Mean Value Theorem, which lead to error estimates for Taylor polynomials. Then we test this generalization on polynomial functions. Recall that the mean value theorem says that, given a continuous function f on a closed interval [a; b], which is differentiable on (a; b), then there is a number c in (a; b) such that f(b) − f(a) f 0(c) = : b − a Rearranging terms, we can make this look very much like the linear approximation for f(b) using the tangent line at x = a: f(b) = f(a) + f 0(c)(b − a) except that the term f 0(a) has been replaced by f 0(c) for some point c in order to achieve an exact equality. Remember that the Mean Value Theorem only gives the existence of such a point c, and not a method for how to find c. We understand this equation as saying that the difference between f(b) and f(a) is given by an expression resembling the next term in the Taylor polynomial. Here f(a) is a \0-th degree" Taylor polynomial. Repeating this for the first degree approximation, we might expect: f 00(c) f(b) = �f(a) + f 0(a)(b − a) � + (b − a)2 2 for some c in (a; b). The term in square brackets is precisely the linear approximation. Question: Guess the formula for the difference between f(b) and its n-th order Taylor polynomial at x = a. Test your answer using the cubic polynomial f(x) = x3 + 2x + 1 using a quadratic approximation for f(3) at x = 1.
    [Show full text]
  • Lecture 5 : Continuous Functions Definition 1 We Say the Function F Is
    Lecture 5 : Continuous Functions Definition 1 We say the function f is continuous at a number a if lim f(x) = f(a): x!a (i.e. we can make the value of f(x) as close as we like to f(a) by taking x sufficiently close to a). Example Last day we saw that if f(x) is a polynomial, then f is continuous at a for any real number a since limx!a f(x) = f(a). If f is defined for all of the points in some interval around a (including a), the definition of continuity means that the graph is continuous in the usual sense of the word, in that we can draw the graph as a continuous line, without lifting our pen from the page. Note that this definition implies that the function f has the following three properties if f is continuous at a: 1. f(a) is defined (a is in the domain of f). 2. limx!a f(x) exists. 3. limx!a f(x) = f(a). (Note that this implies that limx!a− f(x) and limx!a+ f(x) both exist and are equal). Types of Discontinuities If a function f is defined near a (f is defined on an open interval containing a, except possibly at a), we say that f is discontinuous at a (or has a discontinuiuty at a) if f is not continuous at a. This can happen in a number of ways. In the graph below, we have a catalogue of discontinuities. Note that a function is discontinuous at a if at least one of the properties 1-3 above breaks down.
    [Show full text]
  • Week 3 Quiz: Differential Calculus: the Derivative and Rules of Differentiation
    Week 3 Quiz: Differential Calculus: The Derivative and Rules of Differentiation SGPE Summer School 2016 Limits Question 1: Find limx!3f(x): x2 − 9 f(x) = x − 3 (A) +1 (B) -6 (C) 6 (D) Does not exist! (E) None of the above x2−9 (x−3)(x+3) Answer: (C) Note the the function f(x) = x−3 = x−3 = x + 3 is actually a line. However it is important to note the this function is undefined at x = 3. Why? x = 3 requires dividing by zero (which is inadmissible). As x approaches 3 from below and from above, the value of the function f(x) approaches f(3) = 6. Thus the limit limx!3f(x) = 6. Question 2: Find limx!2f(x): f(x) = 1776 (A) +1 (B) 1770 (C) −∞ (D) Does not exist! (E) None of the above Answer: (E) The limit of any constant function at any point, say f(x) = C, where C is an arbitrary constant, is simply C. Thus the correct answer is limx!2f(x) = 1776. Question 3: Find limx!4f(x): f(x) = ax2 + bx + c (A) +1 (B) 16a + 4b + c (C) −∞ (D) Does not exist! (E) None of the above 1 Answer: (B) Applying the rules of limits: 2 2 limx!4ax + bx + c = limx!4ax + limx!4bx + limx!4c 2 = a [limx!4x] + blimx!4x + c = 16a + 4b + c Question 4: Find the limits in each case: (i) lim x2 x!0 jxj (ii) lim 2x+3 x!3 4x−9 (iii) lim x2−3x x!6 x+3 2 Answer: (i) lim x2 = lim (jxj) = lim j x j= 0 x!0 jxj x!0 jxj x!0 (ii) lim 2x+3 = 2·3+3 = 3 x!3 4x−9 4·3−9 (iii) lim x2−3x = 62−3·6 = 2 x!6 x+3 6+3 Question 5: Show that lim sin x = 0 (Hint: −x ≤ sin x ≤ x for all x ≥ 0.) x!0 Answer: Given hint and squeeze theorem we have lim −x = 0 ≤ lim sin x ≤ 0 = lim x hence, x!0 x!0 x!0 lim sin x = 0 x to0 Question 6: Show that lim x sin( 1 ) = 0 x!0 x 1 Answer: Note first that for any real number t we have −1 ≤ sin t ≤ 1 so −1 ≤ sin( x ) ≤ 1.
    [Show full text]