Limit of a function
In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input. Formal definitions, first devised in the early 19th century, are given below. Informally, a function f assigns an output f to every input x. We say the function has a limit L at an input p: this means f gets closer and closer to L as x moves closer and closer to p. More when f is applied to any input sufficiently close to p, the output value is forced arbitrarily close to L. On the other hand, if some inputs close to p are taken to outputs that stay a fixed distance apart, we say the limit does not exist; the notion of a limit has many applications in modern calculus. In particular, the many definitions of continuity employ the limit: a function is continuous if all of its limits agree with the values of the function, it appears in the definition of the derivative: in the calculus of one variable, this is the limiting value of the slope of secant lines to the graph of a function.
Although implicit in the development of calculus of the 17th and 18th centuries, the modern idea of the limit of a function goes back to Bolzano who, in 1817, introduced the basics of the epsilon-delta technique to define continuous functions. However, his work was not known during his lifetime. Cauchy discussed variable quantities and limits and defined continuity of y = f by saying that an infinitesimal change in x produces an infinitesimal change in y in his 1821 book Cours d'analyse, while claims that he only gave a verbal definition. Weierstrass first introduced the epsilon-delta definition of limit in the form it is written today, he introduced the notations lim and limx→x0. The modern notation of placing the arrow below the limit symbol is due to Hardy in his book A Course of Pure Mathematics in 1908. Imagine a person walking over a landscape represented by the graph of y = f, her horizontal position is measured by the value of x, much like the position given by a map of the land or by a global positioning system.
Her altitude is given by the coordinate y. She is walking towards the horizontal position given by x; as she gets closer and closer to it, she notices that her altitude approaches L. If asked about the altitude of x = p, she would answer L. What does it mean to say that her altitude approaches L? It means that her altitude gets nearer and nearer to L except for a possible small error in accuracy. For example, suppose we set a particular accuracy goal for our traveler: she must get within ten meters of L, she reports back that indeed she can get within ten meters of L, since she notes that when she is within fifty horizontal meters of p, her altitude is always ten meters or less from L. The accuracy goal is changed: can she get within one vertical meter? Yes. If she is anywhere within seven horizontal meters of p her altitude always remains within one meter from the target L. In summary, to say that the traveler's altitude approaches L as her horizontal position approaches p means that for every target accuracy goal, however small it may be, there is some neighborhood of p whose altitude fulfills that accuracy goal.
The initial informal statement can now be explicated: The limit of a function f as x approaches p is a number L with the following property: given any target distance from L, there is a distance from p within which the values of f remain within the target distance. This explicit statement is quite close to the formal definition of the limit of a function with values in a topological space. To say that lim x → p f = L, means that ƒ can be made as close as desired to L by making x close enough, but not equal, to p; the following definitions are the accepted ones for the limit of a function in various contexts. Suppose f: R → R is defined on the real line and p,L ∈ R, it is said the limit of f, as x approaches p, is L and written lim x → p f = L, if the following property holds: For every real ε > 0, there exists a real δ > 0 such that for all real x, 0 < | x − p | < δ implies | f − L | < ε. The value of the limit does not depend on the value of f, nor that p be in the domain of f. A more general definition applies for functions defined on subsets of the real line.
Let be an open interval in R, p a point of. Let f be a real-valued function defined on all of except at p itself, it is said that the limit of f, as x approaches p, is L if, for every real ε > 0, there exists a real δ > 0 such that 0 < | x − p | < δ and x ∈ implies | f − L | < ε. Here again the limit does not depend on f being well-defined; the letters ε and δ can be understood as "error" and "distance", in fact Cauchy used ε as an abbreviation for "error" in some of his work, though in his definition of continuity he used an infinitesimal α rather than either ε or δ. In these terms, the error in the measurement of the value at the limit can be made as small as desired by reducing the distance to the limit point; as discussed below this definition works for functions in a more general context. The idea that δ and ε represent distances helps suggest these generalizations. Alternatively x may approach p from
Uncertainty refers to epistemic situations involving imperfect or unknown information. It applies to predictions of future events, to physical measurements that are made, or to the unknown. Uncertainty arises in observable and/or stochastic environments, as well as due to ignorance, indolence, or both, it arises in any number of fields, including insurance, physics, economics, psychology, engineering, meteorology and information science. Although the terms are used in various ways among the general public, many specialists in decision theory and other quantitative fields have defined uncertainty and their measurement as: Uncertainty The lack of certainty, a state of limited knowledge where it is impossible to describe the existing state, a future outcome, or more than one possible outcome. Measurement of uncertainty A set of possible states or outcomes where probabilities are assigned to each possible state or outcome – this includes the application of a probability density function to continuous variables.
Second order uncertainty In statistics and economics, second-order uncertainty is represented in probability density functions over probabilities.. Opinions in subjective logic carry this type of uncertainty. Risk A state of uncertainty where some possible outcomes have an undesired effect or significant loss. Measurement of risk A set of measured uncertainties where some possible outcomes are losses, the magnitudes of those losses – this includes loss functions over continuous variables. Knightian uncertainty In economics, in 1921 Frank Knight distinguished uncertainty from risk with uncertainty being lack of knowledge, immeasurable and impossible to calculate; the essential fact is that'risk' means in some cases a quantity susceptible of measurement, while at other times it is something distinctly not of this character. It will appear that a measurable uncertainty, or'risk' proper, as we shall use the term, is so far different from an unmeasurable one that it is not in effect an uncertainty at all.
Other taxonomies of uncertainties and decisions include a broader sense of uncertainty and how it should be approached from an ethics perspective: For example, if it is unknown whether or not it will rain tomorrow there is a state of uncertainty. If probabilities are applied to the possible outcomes using weather forecasts or just a calibrated probability assessment, the uncertainty has been quantified. Suppose it is quantified as a 90% chance of sunshine. If there is a major, outdoor event planned for tomorrow there is a risk since there is a 10% chance of rain, rain would be undesirable. Furthermore, if this is a business event and $100,000 would be lost if it rains the risk has been quantified; these situations can be made more realistic by quantifying light rain vs. heavy rain, the cost of delays vs. outright cancellation, etc. Some may represent the risk in this example as the "expected opportunity loss" or the chance of the loss multiplied by the amount of the loss; that is useful. Most would be willing to pay a premium to avoid the loss.
An insurance company, for example, would compute an EOL as a minimum for any insurance coverage add onto that other operating costs and profit. Since many people are willing to buy insurance for many reasons clearly the EOL alone is not the perceived value of avoiding the risk. Quantitative uses of the terms uncertainty and risk are consistent from fields such as probability theory, actuarial science, information theory; some create new terms without changing the definitions of uncertainty or risk. For example, surprisal is a variation on uncertainty sometimes used in information theory, but outside of the more mathematical uses of the term, usage may vary widely. In cognitive psychology, uncertainty can be real, or just a matter of perception, such as expectations, etc. Vagueness is a form of uncertainty where the analyst is unable to differentiate between two different classes, such as'person of average height.' and'tall person'. This form of vagueness can be modelled by some variation on Zadeh's fuzzy logic or subjective logic.
Ambiguity is a form of uncertainty where the possible outcomes have unclear meanings and interpretations. The statement "He returns from the bank" is ambiguous because its interpretation depends on whether the word'bank' is meant as "the side of a river" or "a financial institution". Ambiguity arises in situations where multiple analysts or observers have different interpretations of the same statements. Uncertainty may be a consequence of a lack of knowledge of obtainable facts; that is, there may be uncertainty about whether a new rocket design will work, but this uncertainty can be removed with further analysis and experimentation. At the subatomic level, uncertainty may be a unavoidable property of the universe. In quantum mechanics, the Heisenberg uncertainty principle puts limits on how much an observer can know about the position and velocity of a particle; this may not just be ignorance of obtainable facts but that there is no fact to be found. There is some controversy in physics as to whether such uncertainty is an irreducible property of nature or
In iterative reconstruction in digital imaging, interior reconstruction is a technique to correct truncation artifacts caused by limiting image data to a small field of view. The reconstruction focuses on an area known as the region of interest. Although interior reconstruction can be applied to dental or cardiac CT images, the concept is not limited to CT, it is applied with one of several methods. The purpose of each method is to solve for vector x in the following problem: =. Let X be the region of interest and Y be the region outside of X. Assume A, B, C, D are known matrices. X is inside region X, y, in the region Y, is outside region X. F is inside a region in the measurement corresponding to X; this region is denoted as F, while g is outside of the region F. It corresponds to Y and is denoted as G. For CT image-reconstruction purposes, C = 0. To simplify the concept of interior reconstruction, the matrices A, B, C, D are applied to image reconstruction instead of complex operators; the first interior-reconstruction method listed below is extrapolation.
It is a local tomography method which eliminates truncation artifacts but introduces another type of artifact: a bowl effect. An improvement is known as the adaptive extrapolation method, although the iterative extrapolation method below improves reconstruction results. In some cases, the exact reconstruction can be found for the interior reconstruction; the local inverse method below modifies the local tomography method, may improve the reconstruction result of the local tomography. Among the above methods, extrapolation is applied. = A, B, C, D are known matrices. We need to know the vector x. X and y are the original image, while f and g are measurements of responses. Vector x is inside the region of interest X. Vector y is outside the region X; the outside region is called Y, f is inside a region in the measurement corresponding to X. This region is denoted F; the region of vector g corresponds to Y and is denoted as G. In CT image reconstruction, it has C = 0 To simplify the concept of interior reconstruction, the matrices A, B, C, D are applied to image reconstruction instead of a complex operator
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics. By extension, use of complex analysis has applications in engineering fields such as nuclear, aerospace and electrical engineering; as a differentiable function of a complex variable is equal to the sum of its Taylor series, complex analysis is concerned with analytic functions of a complex variable. Complex analysis is one of the classical branches in mathematics, with roots in the 18th century and just prior. Important mathematicians associated with complex numbers include Euler, Riemann, Cauchy and many more in the 20th century. Complex analysis, in particular the theory of conformal mappings, has many physical applications and is used throughout analytic number theory. In modern times, it has become popular through a new boost from complex dynamics and the pictures of fractals produced by iterating holomorphic functions.
Another important application of complex analysis is in string theory which studies conformal invariants in quantum field theory. A complex function is a function from complex numbers to complex numbers. In other words, it is a function that has a subset of the complex numbers as a domain and the complex numbers as a codomain. Complex functions are supposed to have a domain that contains a nonempty open subset of the complex plane. For any complex function, the values z from the domain and their images f in the range may be separated into real and imaginary parts: z = x + i y and f = f = u + i v, where x, y, u, v are all real-valued. In other words, a complex function f: C → C may be decomposed into u: R 2 → R and v: R 2 → R, i.e. into two real-valued functions of two real variables. Any complex-valued function f on an arbitrary set X can be considered as an ordered pair of two real-valued functions: or, alternatively, as a vector-valued function from X into R 2; some properties of complex-valued functions are nothing more than the corresponding properties of vector valued functions of two real variables.
Other concepts of complex analysis, such as differentiability are direct generalizations of the similar concepts for real functions, but may have different properties. In particular, every differentiable complex function is analytic, two differentiable functions that are equal in a neighborhood of a point are equal on the intersection of their domain; the latter property is the basis of the principle of analytic continuation which allows extending every real analytic function in a unique way for getting a complex analytic function whose domain is the whole complex plane with a finite number of curve arcs removed. Many basic and special complex functions are defined in this way, including exponential functions, logarithmic functions, trigonometric functions. Complex functions that are differentiable at every point of an open subset Ω of the complex plane are said to be holomorphic on Ω. In the context of complex analysis, the derivative of f at z 0 is defined to be f ′ = lim z → z 0 f − f z − z 0, z ∈ C.
Superficially, this definition is formally analogous to that of the derivative of a real function. However, complex derivatives and differentiable functions behave in different ways compared to their real counterparts. In particular, for this limit to exist, the value of the difference quotient must approach the same complex number, regardless of the manner in which we
In mathematics, a parabola is a plane curve, mirror-symmetrical and is U-shaped. It fits several superficially different other mathematical descriptions, which can all be proved to define the same curves. One description of a parabola involves a line; the focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane, parallel to another plane, tangential to the conical surface; the line perpendicular to the directrix and passing through the focus is called the "axis of symmetry". The point on the parabola that intersects the axis of symmetry is called the "vertex", is the point where the parabola is most curved; the distance between the vertex and the focus, measured along the axis of symmetry, is the "focal length". The "latus rectum" is the chord of the parabola, parallel to the directrix and passes through the focus.
Parabolas can open up, left, right, or in some other arbitrary direction. Any parabola can be repositioned and rescaled to fit on any other parabola—that is, all parabolas are geometrically similar. Parabolas have the property that, if they are made of material that reflects light light which travels parallel to the axis of symmetry of a parabola and strikes its concave side is reflected to its focus, regardless of where on the parabola the reflection occurs. Conversely, light that originates from a point source at the focus is reflected into a parallel beam, leaving the parabola parallel to the axis of symmetry; the same effects occur with other forms of energy. This reflective property is the basis of many practical uses of parabolas; the parabola has many important applications, from a parabolic antenna or parabolic microphone to automobile headlight reflectors to the design of ballistic missiles. They are used in physics and many other areas; the earliest known work on conic sections was by Menaechmus in the fourth century BC.
He discovered a way to solve the problem of doubling the cube using parabolas. The area enclosed by a parabola and a line segment, the so-called "parabola segment", was computed by Archimedes via the method of exhaustion in the third century BC, in his The Quadrature of the Parabola; the name "parabola" is due to Apollonius. It means "application", referring to "application of areas" concept, that has a connection with this curve, as Apollonius had proved; the focus–directrix property of the parabola and other conic sections is due to Pappus. Galileo showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity; the idea that a parabolic reflector could produce an image was well known before the invention of the reflecting telescope. Designs were proposed in the early to mid seventeenth century by many mathematicians including René Descartes, Marin Mersenne, James Gregory; when Isaac Newton built the first reflecting telescope in 1668, he skipped using a parabolic mirror because of the difficulty of fabrication, opting for a spherical mirror.
Parabolic mirrors are used in most modern reflecting telescopes and in satellite dishes and radar receivers. A parabola can be defined geometrically as a set of points in the Euclidean plane: A parabola is a set of points, such that for any point P of the set the distance | P F | to a fixed point F, the focus, is equal to the distance | P l | to a fixed line l, the directrix: The midpoint V of the perpendicular from the focus F onto the directrix l is called vertex and the line F V the axis of symmetry of the parabola. If one introduces cartesian coordinates, such that F =, f > 0, the directrix has the equation y = − f one obtains for a point P = from | P F | 2 = | P l | 2 the equation x 2 + 2 = 2. Solving for y yields y = 1 4 f x 2; the parabola is U-shaped. The horizontal chord through the focus is called the latus rectum; the latus rectum is parallel to the directrix. The semi-latus
In mathematics, a conic section is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, the ellipse; the circle is a special case of the ellipse, is of sufficient interest in its own right that it was sometimes called a fourth type of conic section. The conic sections have been studied by the ancient Greek mathematicians with this work culminating around 200 BC, when Apollonius of Perga undertook a systematic study of their properties; the conic sections of the Euclidean plane have various distinguishing properties. Many of these have been used as the basis for a definition of the conic sections. One such property defines a non-circular conic to be the set of those points whose distances to some particular point, called a focus, some particular line, called a directrix, are in a fixed ratio, called the eccentricity; the type of conic is determined by the value of the eccentricity. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2.
This equation may be written in matrix form, some geometric properties can be studied as algebraic conditions. In the Euclidean plane, the conic sections appear to be quite different from one another, but share many properties. By extending the geometry to a projective plane this apparent difference vanishes, the commonality becomes evident. Further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically; the conic sections have been studied for thousands of years and have provided a rich source of interesting and beautiful results in Euclidean geometry. A conic is the curve obtained as the intersection of a plane, called the cutting plane, with the surface of a double cone, it shall be assumed that the cone is a right circular cone for the purpose of easy description, but this is not required. Planes that pass through the vertex of the cone will intersect the cone in a point, a line or a pair of intersecting lines; these are called degenerate conics and some authors do not consider them to be conics at all.
Unless otherwise stated, "conic" in this article will refer to a non-degenerate conic. There are three types of conics, the ellipse and hyperbola; the circle is a special kind of ellipse, although it had been considered as a fourth type. The circle and the ellipse arise when the intersection of the plane is a closed curve; the circle is obtained when the cutting plane is parallel to the plane of the generating circle of the cone – for a right cone, see diagram, this means that the cutting plane is perpendicular to the symmetry axis of the cone. If the cutting plane is parallel to one generating line of the cone the conic is unbounded and is called a parabola. In the remaining case, the figure is a hyperbola. In this case, the plane will intersect both halves of the cone, producing two separate unbounded curves. A property that the conic sections share is presented as the following definition. A conic section is the locus of all points P whose distance to a fixed point F is a constant multiple of the distance from P to a fixed line L.
For 0 < e < 1 we obtain an ellipse, for e = 1 a parabola, for e > 1 a hyperbola. A circle is not defined by a focus and directrix, in the plane; the eccentricity of a circle is defined to be zero and its focus is the center of the circle, but there is no line in the Euclidean plane, its directrix. An ellipse and a hyperbola each have distinct directrices for each of them; the line joining the foci is called the principal axis and the points of intersection of the conic with the principal axis are called the vertices of the conic. The line segment joining the vertices of a conic is called the major axis called transverse axis in the hyperbola; the midpoint of this line segment is called the center of the conic. Let a denote the distance from the center to a vertex of an ellipse or hyperbola; the distance from the center to a directrix is a/e while the distance from the center to a focus is ae. A parabola does not have a center; the eccentricity of an ellipse can be seen as a measure of how far the ellipse deviates from being circular.
If the angle between the surface of the cone and its axis is β and the angle between the cutting plane and the axis is α, the eccentricity is cos α cos β. A proof that the conic sections given by the focus-directrix property are the same as those given by planes intersecting a cone is facilitated by the use of Dandelin spheres. Various parameters are associated with a conic section. Recall that the principal axis is the line joining the foci of an ellipse or hyperbola, the center in these cases is the midpoint of the line segment joining the foci; some of the other common features and/or. The linear eccentricity is the distance between the focus; the latus rectum is the chord parallel to the directrix and passing through the focus. Its length is denoted by 2ℓ; the semi-latus rectum is half of the length of the latus rec
Webster's Dictionary is any of the dictionaries edited by Noah Webster in the early nineteenth century, numerous related or unrelated dictionaries that have adopted the Webster's name. "Webster's" has become a genericized trademark in the U. S. for dictionaries of the English language, is used in English dictionary titles. Merriam-Webster is the corporate heir to Noah Webster's original works, which are in the public domain. Noah Webster, the author of the readers and spelling books which dominated the American market at the time, spent decades of research in compiling his dictionaries, his first dictionary, A Compendious Dictionary of the English Language, appeared in 1806. In it, he popularized features which would become a hallmark of American English spelling and included technical terms from the arts and sciences rather than confining his dictionary to literary words. Webster was a proponent of English spelling reform for reasons both nationalistic. In A Companion to the American Revolution, John Algeo notes: "it is assumed that characteristically American spellings were invented by Noah Webster.
He was influential in popularizing certain spellings in America, but he did not originate them. Rather he chose existing options such as center and check on such grounds as simplicity, analogy or etymology". In William Shakespeare's first folios, for example, spellings such as center and color are the most common, he spent the next two decades working to expand his dictionary. In 1828, at the age of 70, Noah Webster published his American Dictionary of the English Language in two quarto volumes containing 70,000 entries, as against the 58,000 of any previous dictionary. There were 2,500 copies printed, at $20 for the two volumes. At first the set sold poorly; when he lowered the price to $15, its sales improved, by 1836 that edition was exhausted. Not all copies were bound at the same time. In 1841, 82-year-old Noah Webster published a second edition of his lexicographical masterpiece with the help of his son, William G. Webster, its title page does not claim the status of second edition noting that this new edition was the "first edition in octavo" in contrast to the quarto format of the first edition of 1828.
Again in two volumes, the title page proclaimed that the Dictionary contained "the whole vocabulary of the quarto, with corrections and several thousand additional words: to, prefixed an introductory dissertation on the origin and connection of the languages of western Asia and Europe, with an explanation of the principles on which languages are formed. B. L. Hamlen of New Haven, prepared the 1841 printing of the second edition; when Webster died, his heirs sold unbound sheets of his 1841 revision American Dictionary of the English Language to the firm of J. S. & C. Adams of Amherst, Massachusetts; this firm bound and published a small number of copies in 1844 – the same edition that Emily Dickinson used as a tool for her poetic composition. However, a $15 price tag on the book made it too expensive to sell so the Amherst firm decided to sell out. Merriam acquired rights from Adams, as well as signing a contract with Webster’s heirs for sole rights; the third printing of the second edition was by George and Charles Merriam of Springfield, Massachusetts, in 1845.
This was the first Webster's Dictionary with a Merriam imprint. Lepore demonstrates Webster's innovative ideas about language and politics and shows why Webster's endeavours were at first so poorly received. Culturally conservative Federalists denounced the work as radical—too inclusive in its lexicon and bordering on vulgar. Meanwhile, Webster's old foes, the Jeffersonian Republicans, attacked the man, labelling him mad for such an undertaking. Scholars have long seen Webster's 1844 dictionary to be an important resource for reading poet Emily Dickinson's life and work. One biographer said, "The dictionary was no mere reference book to her, he shows the ways in which American poetry has inherited Webster and drawn upon his lexicography in order to reinvent it. Austin explicates key definitions from both the Compendious and American dictionaries and brings into its discourse a range of concerns including the politics of American English, the question of national identity and culture in the early moments of American independence, the poetics of citation and of definition.
Webster's dictionaries were a redefinition of Americanism within the context of an emergent and unstable American socio-political and cultural identity. Webster's identification of his project as a "federal language" shows his competing impulses towards regularity and innovation in historical terms; the contradictions of Webster's project represented a part of a larger dialectical play between liberty and order within Revolutionary and post-Revolutionary political debates. Noah Webster's assistant, chief competitor, Joseph Emerson Worcester, Webster's son-in-law Chauncey A. Goodrich, published an abridgment of Noah Webster's 1828 American Dictionary of the English Language in 1829, with the same number of words and Webster's full definitions, but with truncated literary references and expanded etymology. Although it was more successful f