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Rutherford model
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Basic diagram of the atomic nuclear model: electrons in green and nucleus in red
Play media
3D animation of an atom incorporating the Rutherford model
The Rutherford model is a model of the atom devised by Ernest Rutherford. Rutherford directed the famous Geiger–Marsden experiment in 1909 which suggested, upon Rutherford's 1911 analysis, that J. J. Thomson's plum pudding model of the atom was incorrect. Rutherford's new model[1] for the atom, based on the experimental results, contained the new features of a relatively high central charge concentrated into a very small volume in comparison to the rest of the atom and with this central volume also containing the bulk of the atomic mass of the atom. This region would be known as the "nucleus" of the atom.
Contents
[hide]
1 Experimental basis for the model
2 Contribution to modern science
3 Symbolism
4 References
5 External links
Experimental basis for the model[edit]
Rutherford overturned Thomson's model in 1911 with his well-known gold foil experiment in which he demonstrated that the atom has a tiny and heavy nucleus. Rutherford designed an experiment to use the alpha particles emitted by a radioactive element as probes to the unseen world of atomic structure. If Thomson was correct, the beam would go straight through the gold foil. Most of the beams went through the foil, but a few were reflected.
Rutherford presented his own physical model for subatomic structure, as an interpretation for the unexpected experimental results. In it, the atom is made up of a central charge (this is the modern atomic nucleus, though Rutherford did not use the term "nucleus" in his paper) surrounded by a cloud of (presumably) orbiting electrons. In this May 1911 paper, Rutherford only committed himself to a small central region of very high positive or negative charge in the atom.
For concreteness, consider the passage of a high speed α particle through an atom having a positive central charge N e, and surrounded by a compensating charge of N electrons.[2]
From purely energetic considerations of how far particles of known speed would be able to penetrate toward a central charge of 100 e, Rutherford was able to calculate that the radius of his gold central charge would need to be less (how much less could not be told) than 3.4 × 10−14 meters. This was in a gold atom known to be 10−10 meters or so in radius—a very surprising finding, as it implied a strong central charge less than 1/3000th of the diameter of the atom.
The Rutherford model served to concentrate a great deal of the atom's charge and mass to a very small core, but didn't attribute any structure to the remaining electrons and remaining atomic mass. It did mention the atomic model of Hantaro Nagaoka, in which the electrons are arranged in one or more rings, with the specific metaphorical structure of the stable rings of Saturn. The plum pudding model of J. J. Thomson also had rings of orbiting electrons. Jean Baptiste Perrin claimed in his Nobel lecture[3] that he was the first one to suggest the model in his paper dated 1901.
The Rutherford paper suggested that the central charge of an atom might be "proportional" to its atomic mass in hydrogen mass units u (roughly 1/2 of it, in Rutherford's model). For gold, this mass number is 197 (not then known to great accuracy) and was therefore modeled by Rutherford to be possibly 196 u. However, Rutherford did not attempt to make the direct connection of central charge to atomic number, since gold's "atomic number" (at that time merely its place number in the periodic table) was 79, and Rutherford had modeled the charge to be about +100 units (he had actually suggested 98 units of positive charge, to make half of 196). Thus, Rutherford did not formally suggest the two numbers (periodic table place, 79, and nuclear charge, 98 or 100) might be exactly the same.
A month after Rutherford's paper appeared, the proposal regarding the exact identity of atomic number and nuclear charge was made by Antonius van den Broek, and later confirmed experimentally within two years, by Henry Moseley.
These are the key indicators-
The atom's electron cloud does not influence alpha particle scattering.
Much of an atom's positive charge is concentrated in a relatively tiny volume at the center of the atom, known today as the nucleus. The magnitude of this charge is proportional to (up to a charge number that can be approximately half of) the atom's atomic mass—the remaining mass is now known to be mostly attributed to neutrons. This concentrated central mass and charge is responsible for deflecting both alpha and beta particles.
The mass of heavy atoms such as gold is mostly concentrated in the central charge region, since calculations show it is not deflected or moved by the high speed alpha particles, which have very high momentum in comparison to electrons, but not with regard to a heavy atom as a whole.
The atom itself is about 100,000 (105) times the diameter of the nucleus.[4] This could be related to putting a grain of sand in the middle of a football field.[5]
Contribution to modern science[edit]
After Rutherford's discovery, scientists started to realize that the atom is not ultimately a single particle, but is made up of far smaller subatomic particles. Subsequent research determined the exact atomic structure which led to Rutherford's gold foil experiment. Scientists eventually discovered that atoms have a positively charged nucleus (with an exact atomic number of charges) in the center, with a radius of about 1.2 × 10−15 meters × [Atomic Mass Number] 1⁄3. Electrons were found to be even smaller.
Later, scientists found the expected number of electrons (the same as the atomic number) in an atom by using X-rays. When an X-ray passes through an atom, some of it is scattered while the rest passes through the atom. Since the X-ray loses its intensity primarily due to scattering at electrons, by noting the rate of decrease in X-ray intensity, the number of electrons contained in an atom can be accurately estimated.
Symbolism[edit]
See also Bohr model, which applies just as well to the section below.
Generic atomic planetary model.
Shield of the U.S. Atomic Energy Commission
Rutherford's model deferred to the idea of many electrons in rings, per Nagaoka. However, once Niels Bohr modified this view into a picture of just a few planet-like electrons for light atoms, the Rutherford-Bohr model caught the imagination of the public. It has since continually been used as a symbol for atoms and even for "atomic" energy (even though this is more properly considered nuclear energy). Examples of its use over the past century include but are not limited to:
The logo of the United States Atomic Energy Commission, which was in part responsible for its later usage in relation to nuclear fission technology in particular.
The flag of the International Atomic Energy Agency is a Rutherford atom, enclosed in olive branches.
The US minor league baseball Albuquerque Isotopes' logo is a Rutherford atom, with the electron orbits forming an A.
A similar symbol, the Atomic whirl, was chosen as the symbol for the American Atheists, and has come to be used as a symbol of atheism in general.
The Unicode Miscellaneous Symbols codepoint U+269B (⚛) uses a Rutherford atom.
On maps, it is generally used to indicate a nuclear power installation.
References[edit]
Jump up ^ Akhlesh Lakhtakia (Ed.); Salpeter, Edwin Ε. (1996). "Models and Modelers of Hydrogen". American Journal of Physics. World Scientific. 65 (9): 933. Bibcode:1997AmJPh..65..933L. doi:10.1119/1.18691. ISBN 981-02-2302-1. CS1 maint: Extra text: authors list (link)
Jump up ^ E. Rutherford, "The Scattering of α and β Particles by Matter and the Structure of the Atom", Philosophical Magazine. Series 6, vol. 21. May 1911
Jump up ^ 1926 Lecture for Nobel Prize in Physics
Jump up ^ Nicholas Giordano (1 January 2012). College Physics: Reasoning and Relationships. Cengage Learning. pp. 1051–. ISBN 1-285-22534-1.
Jump up ^ Constan, Zach (2010). "Learning Nuclear Science with Marbles". The Physics Teacher. 48 (2): 114. Bibcode:2010PhTea..48..114C. doi:10.1119/1.3293660.
External links[edit]
Rutherford's Model by Raymond College
Rutherford's Model by Kyushu University
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Theatre technique
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Theatre techniques are procedures that facilitate a successful presentation of a play. They also include any practices that advance and enhance the understanding the audience brings to the action and the acting by the cast on stage.
Contents
[hide]
1 The playwright's craft
2 The director’s craft
3 Stage management and stagecraft
4 Trends and movements
4.1 The three unities
5 Theatre presentation
5.1 Defamiliarization effect (Verfremdungseffekt)
6 Teichoscopy
7 See also
The playwright's craft[edit]
Theatre technique is part of the playwright's creative writing of drama, as a kind of mimesis rather than mere illusion or imitation of life, in that the playwright is able to present a reality to the audience that is different, yet recognisable to that which they usually identify with in their everyday lives.
Another aspect of this is that of creating the kind of dialogue that makes the playwright's characters come alive and allows for their development in the course of his dramatization.
The playwright's art also consists in the ability to convey to the audience the ideas that give essence to the drama within the frame of its structure. Finally, the feeling for the natural divisions of a play—including acts, scenes, and changes of place—its entries and exits, and the positioning of the cast are integral to playwriting technique.
One of the playwright’s functions is that concerned with adaptations of existing traditional drama, such as Charles Marowitz’s collages of Hamlet and Macbeth and other re-interpretations of Shakespeare's works, as well as Tom Stoppard’s approaches in Rosencrantz & Guildenstern Are Dead, Dogg's Hamlet, and Cahoot's Macbeth.
The director’s craft[edit]
The director these days is responsible for the actual production of a play, as opposed to earlier days when it was the producer who, at least in Britain until the 1950s, had this task. In earlier centuries it was the author, an actor-manager, or a leading actor with whom the responsibilities of staging a drama were invested.
The director produces the play in the way he envisages how it ought to be seen as he interprets what the playwright intended within the drama; he takes care of the effictiveness of the rehearsals of the actors; and coordinates the work of designers and technicians in the production.
However, the playwright’s work is still reflected in the director’s prompt copy, a separate form of stage instructions worked out in detail by the director, in which each actor is given details as to what is happening onstage, where exactly he has to be in relation to the back, front, left, or right of the stage, and what he is to do at any one time during the play.
Stage management and stagecraft[edit]
The stage manager has to work with the director, and ascertain whether both the director’s objectives and the perceptions of the stage designers are compatible and realistic. He is usually the link between the director and the rest of the company, and is responsible for the director’s visions being passed down to each actor and member of the running crew. He is also responsible for safety and running an orderly backstage area. He maintains the prompt book and the call board, to which rehearsal schedule notes are affixed for the cast.
Stagecraft overall has to address the various shortcomings of the stage’s spatial and physical limitations. For example, the stage alone cannot be expected to provide wide and distant vistas, or vast spaces where armies gather or huge masses of people congregate. Natural phenomena such as thunderstorms or winds, that are often part of the drama, cannot be recreated in their original form. Furthermore, because of the distance involved, it is difficult for actors to portray the feelings, tensions and passions of their characters to the audience with much distinctiveness.
It is for these reasons that special technologies and techniques have been developed from classical times onward to supplement and augment the effects that are to be realised. Technical specialists help to implement these techniques by providing expertise in various areas of the production. The stage manager and his team must ultimately draw all these separate techniques together to create an effective and successful stage production.
Sets and sceneries have to be designed and created in the workshop. They include outdoor and indoor scenes, or special types of constructions, and must also usually achieve the appearance of depth and distance within the spatial limitations of the stage.
Stage lighting has to be provided to set the moods and sensitivities of the play, and to permit the showing of the emotional charge in the face of the actors, possibly reinforcing the pallor or vividness of their faces.
Costumes have to typify characters from the moment they appear on stage and signify the period or the social milieu in which the characters are to be seen. They can also indicate the characters' circumstances (whether they should appear rich or poor) or even whether they should be seen as comic or tragic personae.
Sound effects have to convey natural phenomena, such as wind or storm. However, it is also their function to call forth moods and feelings the audience will recognise.
Special effects have to deliver the imitations of physical actions on stage, such as explosions, fireworks, fog, or even earthquakes that are often part of the story.
Rigging is an important aspect that is not often visible. All sound and lighting features and their cabling and wiring have to be held in a safe position, above both the stage and the auditorium, and therefore require specialist temporary and permanent fittings to be installed.
The technical director, especially in larger theatres, has the responsibility of overseeing the rigging and construction of the stage scenery. He creates working drawings from the scenic designer’s drawings in order to pass on the designs to the scene shop.
Trends and movements[edit]
The three unities[edit]
The Classical unities of time, action and place were the main principles of French neo-classical drama during part of the 17th century.
They were introduced by Jean Mairet after a misreading of Aristotle's Poetics, and the critic Castelvetro insisted that playwrights and directors adhere to the unities. In the Poetics Aristotle had merely recommended that action should consist only of the main plot without any subplots, and that the time represented by action should not stretch beyond the length of one day. Time merely entered into his recommendations as a hint as to the limits of the attention span the audience could be expected to have. The unity of place, that of the confinement of the action in a play to one locality only, was not mentioned at all.
The effect of the three unities on French drama during this period was that their presentation became very restrictive, and it was only when later dramatists began to avoid mentioning specific times and places that the presentation of plays became more creative again.
Theatre presentation[edit]
Some dramatists and dramaturgists try to achieve particular effects that are not normally sought in a theatre presentation.
Defamiliarization effect (Verfremdungseffekt)[edit]
Bertolt Brecht coined the term "defamiliarization effect" (sometimes called "estrangement effect" or "alienation effect"; German Verfremdungseffekt) for an approach to theater that focused on the central ideas and decisions in the play, and discouraged involving the audience in an illusory world and in the emotions of the characters. Brecht thought the audience required an emotional distance to reflect on what is being presented. See epic theater.
Teichoscopy[edit]
One of the oldest techniques that has been used often, is that of teichoscopy or the "viewing from the wall", in which actors observe events beyond the confines of the stage, such as a distant battle, and discuss it on stage while the battle is taking place, as opposed to the event being reported by messengers at a later time after the event has happened. Shakespeare uses this technique in the final scenes of the play Julius CeasreJulius Caesar.
See also[edit]
Literary technique
Classical unities
Theatre
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Limit of a function
From Wikipedia, the free encyclopedia
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x
sin  x x {\displaystyle {\frac {\sin x}{x}}}
1 0.841471...
0.1 0.998334...
0.01 0.999983...
Although the function (sin x)/x is not defined at zero, as x becomes closer and closer to zero, (sin x)/x becomes arbitrarily close to 1. In other words, the limit of (sin x)/x as x approaches zero equals 1.
Part of a series of articles about
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Glossary of calculus

v
t
e
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(x) to every input x. We say the function has a limit L at an input p: this means f(x) gets closer and closer to L as x moves closer and closer to p. More specifically, 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 very 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: roughly, a function is continuous if all of its limits agree with the values of the function. It also 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.
Contents
[hide]
1 History
2 Motivation
3 Functions of a single variable
3.1 Existence and one-sided limits
3.2 More general subsets
3.3 Deleted versus non-deleted limits
3.4 Examples
3.4.1 Non-existence of one-sided limit(s)
3.4.2 Non-equality of one-sided limits
3.4.3 Limits at only one point
3.4.4 Limits at countably many points
4 Functions on metric spaces
5 Functions on topological spaces
6 Limits involving infinity
6.1 Limits at infinity
6.2 Infinite limits
6.3 Alternative notation
6.4 Limits at infinity for rational functions
7 Functions of more than one variable
8 Sequential limits
9 Other characterizations
9.1 In terms of sequences
9.2 In non-standard calculus
9.3 In terms of nearness
10 Relationship to continuity
11 Properties
11.1 Limits of compositions of functions
11.2 Limits of special interest
11.2.1 Rational functions
11.2.2 Trigonometric functions
11.2.3 Exponential functions
11.2.4 Logarithmic functions
11.3 L'Hôpital's rule
11.4 Summations and integrals
12 See also
13 Notes
14 References
History[edit]
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 (Felscher 2000). Cauchy discussed variable quantities, infinitesimals, and limits and defined continuity of y = f ( x ) {\displaystyle y=f(x)} by saying that an infinitesimal change in x necessarily produces an infinitesimal change in y in his 1821 book Cours d'analyse, while (Grabiner 1983) claims that he only gave a verbal definition. Weierstrass first introduced the epsilon-delta definition of limit in the form it is usually written today. He also introduced the notations lim and limx→x0 (Burton 1997).
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 (Miller 2004).
Motivation[edit]
Imagine a person walking over a landscape represented by the graph of y = f(x). 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 = p. 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 then answer L.
What, then, 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 then changed: can she get within one vertical meter? Yes. If she is anywhere within seven horizontal meters of p, then 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(x) 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(x) 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 ( x ) = L , {\displaystyle \lim _{x\to p}f(x)=L,\,}
means that ƒ(x) can be made as close as desired to L by making x close enough, but not equal, to p.
The following definitions (known as (ε, δ)-definitions) are the generally accepted ones for the limit of a function in various contexts.
Functions of a single variable[edit]
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 ( x ) = L , {\displaystyle \lim _{x\to p}f(x)=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(x) − L | < ε.
The value of the limit does not depend on the value of f(p), nor even that p be in the domain of f.
A more general definition applies for functions defined on subsets of the real line. Let (a, b) be an open interval in R, and p a point of (a, b). Let f be a real-valued function defined on all of (a, b) except possibly at p itself. It is then 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 ∈ (a, b) implies | f(x) − L | < ε.
Here again the limit does not depend on f(p) being well-defined.
The letters ε and δ can be understood as "error" and "distance", and in fact Cauchy used ε as an abbreviation for "error" in some of his work (Grabiner 1983), though in his definition of continuity he used an infinitesimal α {\displaystyle \alpha } rather than either ε or δ (see Cours d'Analyse). In these terms, the error (ε) in the measurement of the value at the limit can be made as small as desired by reducing the distance (δ) to the limit point. As discussed below this definition also works for functions in a more general context. The idea that δ and ε represent distances helps suggest these generalizations.
Existence and one-sided limits[edit]
Main article: One-sided limit
The limit as: x → x0+ ≠ x → x0−. Therefore, the limit as x → x0 does not exist.
Alternatively x may approach p from above (right) or below (left), in which case the limits may be written as
lim x → p + f ( x ) = L {\displaystyle \lim _{x\to p^{+}}f(x)=L}
or
lim x → p − f ( x ) = L {\displaystyle \lim _{x\to p^{-}}f(x)=L}
respectively. If these limits exist at p and are equal there, then this can be referred to as the limit of f(x) at p. If the one-sided limits exist at p, but are unequal, there is no limit at p (the limit at p does not exist). If either one-sided limit does not exist at p, the limit at p does not exist.
A formal definition is as follows. The limit of f(x) as x approaches p from above is L if, for every ε > 0, there exists a δ > 0 such that |f(x) − L| < ε whenever 0 < x − p < δ. The limit of f(x) as x approaches p from below is L if, for every ε > 0, there exists a δ > 0 such that |f(x) − L| < ε whenever 0 < p − x < δ.
If the limit does not exist then the oscillation of f at p is non-zero.
More general subsets[edit]
Apart from open intervals, limits can be defined for functions on arbitrary subsets of R, as follows. Let f be a real-valued function defined on a subset S of the real line. Let p be a limit point of S—that is, p is the limit of some sequence of distinct elements of S. The limit of f, as x approaches p from values in S, is L if, for every ε > 0, there exists a δ > 0 such that 0 < |x − p| < δ and x ∈ S implies |f(x) − L| < ε.
This limit is often written
L = lim x → p x ∈ S f ( x ) . {\displaystyle L={\underset {x\in S}{\lim _{x\to p}}}f(x).}
The condition that f be defined on S is that S be a subset of the domain of f. This generalization includes as special cases limits on an interval, as well as left-handed limits of real-valued functions (e.g., by taking S to be an open interval of the form ( − ∞ , a ) {\displaystyle (-\infty ,a)} ), and right-handed limits (e.g., by taking S to be an open interval of the form ( a , ∞ ) {\displaystyle (a,\infty )} ).
Deleted versus non-deleted limits[edit]
The definition of limit given here does not depend on how (or whether) f is defined at p. Bartle (1967), refers to this as a deleted limit, because it excludes the value of f at p. The corresponding non-deleted limit does depend on the value of f at p, if p is in the domain of f:
A number L is the non-deleted limit of f as x approaches p if, for every ε > 0, there exists a δ > 0 such that | x − p | < δ and x ∈ Dm(f) implies | f(x) − L | < ε.
The definition is the same, except that the neighborhood | x − p | < δ now includes the point p, in contrast to the (deleted) neighborhood 0 < | x − p | < δ. This makes the definition of a non-deleted limit less general. One of the advantages of working with non-deleted limits is that they allow to state the theorem about limits of compositions without any constraints on the functions (other than the existence of their non-deleted limits) (Hubbard (2015)).
Bartle (1967) notes that although by "limit" some authors do mean this non-deleted limit, deleted limits are the most popular. For example, Apostol (1974), Courant (1924), Hardy (1921), Rudin (1964), Whittaker & Watson (1902) all by "limit" mean the deleted version.
Examples[edit]
Non-existence of one-sided limit(s)[edit]
The function without a limit, at an essential discontinuity
The function
f ( x ) = { sin  5 x − 1 for x < 1 0 for x = 1 0.1 x − 1 for x > 1 {\displaystyle f(x)={\begin{cases}\sin {\frac {5}{x-1}}&{\text{ for }}x<1\\0&{\text{ for }}x=1\\{\frac {0.1}{x-1}}&{\text{ for }}x>1\end{cases}}}
has no limit at x 0 = 1 {\displaystyle x_{0}=1} (the left-hand limit does not exist due to the oscillatory nature of the sine function, and the right-hand limit does not exist due to the asymptotic behaviour of the reciprocal function), but has a limit at every other x-coordinate.
The function
f ( x ) = { 1 x rational 0 x irrational {\displaystyle f(x)={\begin{cases}1&x{\text{ rational }}\\0&x{\text{ irrational }}\end{cases}}}
(the Dirichlet function) has no limit at any x-coordinate.
Non-equality of one-sided limits[edit]
The function
f ( x ) = { 1 for x < 0 2 for x ≥ 0 {\displaystyle f(x)={\begin{cases}1&{\text{ for }}x<0\\2&{\text{ for }}x\geq 0\end{cases}}}
has a limit at every non-zero x-coordinate (the limit equals 1 for negative x and equals 2 for positive x). The limit at x = 0 does not exist (the left-hand limit equals 1, whereas the right-hand limit equals 2).
Limits at only one point[edit]
The functions
f ( x ) = { x x rational 0 x irrational {\displaystyle f(x)={\begin{cases}x&x{\text{ rational }}\\0&x{\text{ irrational }}\end{cases}}}
and
f ( x ) = { | x | x rational 0 x irrational {\displaystyle f(x)={\begin{cases}|x|&x{\text{ rational }}\\0&x{\text{ irrational }}\end{cases}}}
both have a limit at x = 0 and it equals 0.
Limits at countably many points[edit]
The function
f ( x ) = { sin  x x irrational 1 x rational {\displaystyle f(x)={\begin{cases}\sin x&x{\text{ irrational }}\\1&x{\text{ rational }}\end{cases}}}
has a limit at any x-coordinate of the form π 2 + 2 n π {\displaystyle {\frac {\pi }{2}}+2n\pi } , where n is any integer.
Functions on metric spaces[edit]
Suppose M and N are subsets of metric spaces A and B, respectively, and f : M → N is defined between M and N, with x ∈ M, p a limit point of M and L ∈ N. It is said that the limit of f as x approaches p is L and write
lim x → p f ( x ) = L {\displaystyle \lim _{x\to p}f(x)=L}
if the following property holds:
For every ε > 0, there exists a δ > 0 such that dB(f(x), L) < ε whenever 0 < dA(x, p) < δ.
Again, note that p need not be in the domain of f, nor does L need to be in the range of f, and even if f(p) is defined it need not be equal to L.
An alternative definition using the concept of neighbourhood is as follows:
lim x → p f ( x ) = L {\displaystyle \lim _{x\to p}f(x)=L}
if, for every neighbourhood V of L in B, there exists a neighbourhood U of p in A such that f(U ∩ M − {p}) ⊆ V.
Functions on topological spaces[edit]
Suppose X,Y are topological spaces with Y a Hausdorff space. Let p be a limit point of Ω ⊆ X, and L ∈Y. For a function f : Ω → Y, it is said that the limit of f as x approaches p is L (i.e., f(x) → L as x → p) and written
lim x → p f ( x ) = L {\displaystyle \lim _{x\to p}f(x)=L}
if the following property holds:
For every open neighborhood V of L, there exists an open neighborhood U of p such that f(U ∩ Ω − {p}) ⊆ V.
This last part of the definition can also be phrased "there exists an open punctured neighbourhood U of p such that f(U∩Ω) ⊆ V ".
Note that the domain of f does not need to contain p. If it does, then the value of f at p is irrelevant to the definition of the limit. In particular, if the domain of f is X − {p} (or all of X), then the limit of f as x → p exists and is equal to L if, for all subsets Ω of X with limit point p, the limit of the restriction of f to Ω exists and is equal to L. Sometimes this criterion is used to establish the non-existence of the two-sided limit of a function on R by showing that the one-sided limits either fail to exist or do not agree. Such a view is fundamental in the field of general topology, where limits and continuity at a point are defined in terms of special families of subsets, called filters, or generalized sequences known as nets.
Alternatively, the requirement that Y be a Hausdorff space can be relaxed to the assumption that Y be a general topological space, but then the limit of a function may not be unique. In particular, one can no longer talk about the limit of a function at a point, but rather a limit or the set of limits at a point.
A function is continuous in a limit point p of and in its domain if and only if f(p) is the (or, in the general case, a) limit of f(x) as x tends to p.
Limits involving infinity[edit]
Limits at infinity[edit]
The limit of this function at infinity exists.
For f(x) a real function, the limit of f as x approaches infinity is L, denoted
lim x → ∞ f ( x ) = L , {\displaystyle \lim _{x\to \infty }f(x)=L,}
means that for all ε > 0 {\displaystyle \varepsilon >0} , there exists c such that | f ( x ) − L | < ε {\displaystyle |f(x)-L|<\varepsilon } whenever x > c. Or, symbolically:
∀ ε > 0 ∃ c ∀ x > c : | f ( x ) − L | < ε {\displaystyle \forall \varepsilon >0\;\exists c\;\forall x>c:\;|f(x)-L|<\varepsilon } .
Similarly, the limit of f as x approaches negative infinity is L, denoted
lim x → − ∞ f ( x ) = L , {\displaystyle \lim _{x\to -\infty }f(x)=L,}
means that for all ε > 0 {\displaystyle \varepsilon >0} there exists c such that | f ( x ) − L | < ε {\displaystyle |f(x)-L|<\varepsilon } whenever x < c. Or, symbolically:
∀ ε > 0 ∃ c ∀ x < c : | f ( x ) − L | < ε {\displaystyle \forall \varepsilon >0\;\exists c\;\forall x<c:\;|f(x)-L|<\varepsilon } .
For example
lim x → − ∞ e x = 0. {\displaystyle \lim _{x\to -\infty }e^{x}=0.\,}
Infinite limits[edit]
Limits can also have infinite values. When infinities are not considered legitimate values, which is standard (but see below), a formalist will insist upon various circumlocutions. For example, rather than say that a limit is infinity, the proper thing is to say that the function "diverges" or "grows without bound". In particular, the following informal example of how to pronounce the notation is arguably inappropriate in the classroom (or any other formal setting). In any case, for example the limit of f as x approaches a is infinity, denoted
lim x → a f ( x ) = ∞ , {\displaystyle \lim _{x\to a}f(x)=\infty ,\,}
means that for all ε > 0 {\displaystyle \varepsilon >0} there exists δ > 0 {\displaystyle \delta >0} such that f ( x ) > ε {\displaystyle f(x)>\varepsilon } whenever | x − a | < δ {\displaystyle |x-a|<\delta } .
These ideas can be combined in a natural way to produce definitions for different combinations, such as
lim x → ∞ f ( x ) = ∞ , lim x → a + f ( x ) = − ∞ . {\displaystyle \lim _{x\to \infty }f(x)=\infty ,\lim _{x\to a^{+}}f(x)=-\infty .\,}
For example
lim x → 0 + ln  x = − ∞ . {\displaystyle \lim _{x\to 0^{+}}\ln x=-\infty .\,}
Limits involving infinity are connected with the concept of asymptotes.
These notions of a limit attempt to provide a metric space interpretation to limits at infinity. However, note that these notions of a limit are consistent with the topological space definition of limit if
a neighborhood of −∞ is defined to contain an interval [−∞, c) for some c ∈ R
a neighborhood of ∞ is defined to contain an interval (c, ∞] where c ∈ R
a neighborhood of a∈R is defined in the normal way metric space R
In this case, R is a topological space and any function of the form f: X → Y with X, Y⊆ R is subject to the topological definition of a limit. Note that with this topological definition, it is easy to define infinite limits at finite points, which have not been defined above in the metric sense.
Alternative notation[edit]
Many authors[1] allow for the projectively extended real line to be used as a way to include infinite values as well as extended real line. With this notation, the extended real line is given as R ∪ {−∞, +∞} and the projectively extended real line is R ∪ {∞} where a neighborhood of ∞ is a set of the form {x: |x| > c}. The advantage is that one only needs three definitions for limits (left, right, and central) to cover all the cases. As presented above, for a completely rigorous account, we would need to consider 15 separate cases for each combination of infinities (five directions: −∞, left, central, right, and +∞; three bounds: −∞, finite, or +∞). There are also noteworthy pitfalls. For example, when working with the extended real line, x − 1 {\displaystyle x^{-1}} does not possess a central limit (which is normal):
lim x → 0 + 1 x = + ∞ , lim x → 0 − 1 x = − ∞ . {\displaystyle \lim _{x\to 0^{+}}{1 \over x}=+\infty ,\lim _{x\to 0^{-}}{1 \over x}=-\infty .}
In contrast, when working with the projective real line, infinities (much like 0) are unsigned, so, the central limit does exist in that context:
lim x → 0 + 1 x = lim x → 0 − 1 x = lim x → 0 1 x = ∞ . {\displaystyle \lim _{x\to 0^{+}}{1 \over x}=\lim _{x\to 0^{-}}{1 \over x}=\lim _{x\to 0}{1 \over x}=\infty .}
In fact there are a plethora of conflicting formal systems in use. In certain applications of numerical differentiation and integration, it is, for example, convenient to have signed zeroes. A simple reason has to do with the converse of lim x → 0 − x − 1 = − ∞ {\displaystyle \lim _{x\to 0^{-}}{x^{-1}}=-\infty } , namely, it is convenient for lim x → − ∞ x − 1 = − 0 {\displaystyle \lim _{x\to -\infty }{x^{-1}}=-0} to be considered true. Such zeroes can be seen as an approximation to infinitesimals.
Limits at infinity for rational functions[edit]
Horizontal asymptote about y = 4
There are three basic rules for evaluating limits at infinity for a rational function f(x) = p(x)/q(x): (where p and q are polynomials):
If the degree of p is greater than the degree of q, then the limit is positive or negative infinity depending on the signs of the leading coefficients;
If the degree of p and q are equal, the limit is the leading coefficient of p divided by the leading coefficient of q;
If the degree of p is less than the degree of q, the limit is 0.
If the limit at infinity exists, it represents a horizontal asymptote at y = L. Polynomials do not have horizontal asymptotes; such asymptotes may however occur with rational functions.
Functions of more than one variable[edit]
By noting that |x − p| represents a distance, the definition of a limit can be extended to functions of more than one variable. In the case of a function f : R2 → R,
lim ( x , y ) → ( p , q ) f ( x , y ) = L {\displaystyle \lim _{(x,y)\to (p,q)}f(x,y)=L}
if
for every ε > 0 there exists a δ > 0 such that for all (x,y) with 0 < ||(x,y) − (p,q)|| < δ, then |f(x,y) − L| < ε
where ||(x,y) − (p,q)|| represents the Euclidean distance. This can be extended to any number of variables.
Sequential limits[edit]
Let f : X → Y be a mapping from a topological space X into a Hausdorff space Y, p ∈ X and L ∈ Y.
The sequential limit of f as x tends to p is L if, for every sequence (xn) in X − {p} that converges to p, the sequence f(xn) converges to L.
If L is the limit (in the sense above) of f as x approaches p, then it is a sequential limit as well, however the converse need not hold in general. If in addition X is metrizable, then L is the sequential limit of f as x approaches p if and only if it is the limit (in the sense above) of f as x approaches p.
Other characterizations[edit]
In terms of sequences[edit]
For functions on the real line, one way to define the limit of a function is in terms of the limit of sequences. (This definition is usually attributed to Eduard Heine.) In this setting:
lim x → a f ( x ) = L {\displaystyle \lim _{x\to a}f(x)=L}
if and only if for all sequences x n {\displaystyle x_{n}} (with x n {\displaystyle x_{n}} not equal to a for all n) converging to a {\displaystyle a} the sequence f ( x n ) {\displaystyle f(x_{n})} converges to L {\displaystyle L} . It was shown by Sierpiński in 1916 that proving the equivalence of this definition and the definition above, requires and is equivalent to a weak form of the axiom of choice. Note that defining what it means for a sequence x n {\displaystyle x_{n}} to converge to a {\displaystyle a} requires the epsilon, delta method.
Similarly as it was the case of Weierstrass's definition, a more general Heine definition applies to functions defined on subsets of the real line. Let f be a real-valued function with the domain Dm(f). Let a be the limit of a sequence of elements of Dm(f). Then the limit (in this sense) of f is L as x approaches p if for every sequence x n {\displaystyle x_{n}} ∈ Dm(f) \ {a} (so that for all n, x n {\displaystyle x_{n}} is not equal to a) that converges to a, the sequence f ( x n ) {\displaystyle f(x_{n})} converges to L {\displaystyle L} . This is the same as the definition of a sequential limit in the preceding section obtained by regarding the subset Dm(f) of R as a metric space with the induced metric.
In non-standard calculus[edit]
In non-standard calculus the limit of a function is defined by:
lim x → a f ( x ) = L {\displaystyle \lim _{x\to a}f(x)=L}
if and only if for all x ∈ R ∗ {\displaystyle x\in \mathbb {R} ^{*}} , f ∗ ( x ) − L {\displaystyle f^{*}(x)-L} is infinitesimal whenever x − a {\displaystyle x-a} is infinitesimal. Here R ∗ {\displaystyle \mathbb {R} ^{*}} are the hyperreal numbers and f ∗ {\displaystyle f^{*}} is the natural extension of f to the non-standard real numbers. Keisler proved that such a hyperreal definition of limit reduces the quantifier complexity by two quantifiers.[2] On the other hand, Hrbacek writes that for the definitions to be valid for all hyperreal numbers they must implicitly be grounded in the ε-δ method, and claims that, from the pedagogical point of view, the hope that non-standard calculus could be done without ε-δ methods cannot be realized in full.[3] Bŀaszczyk et al. detail the usefulness of microcontinuity in developing a transparent definition of uniform continuity, and characterize Hrbacek's criticism as a "dubious lament".[4]
In terms of nearness[edit]
At the 1908 international congress of mathematics F. Riesz introduced an alternate way defining limits and continuity in concept called "nearness". A point x {\displaystyle x} is defined to be near a set A ⊆ R {\displaystyle A\subseteq \mathbb {R} } if for every r > 0 {\displaystyle r>0} there is a point a ∈ A {\displaystyle a\in A} so that | x − a | < r {\displaystyle |x-a|<r} . In this setting the
lim x → a f ( x ) = L {\displaystyle \lim _{x\to a}f(x)=L}
if and only if for all A ⊆ R {\displaystyle A\subseteq \mathbb {R} } , L {\displaystyle L} is near f ( A ) {\displaystyle f(A)} whenever a {\displaystyle a} is near A {\displaystyle A} . Here f ( A ) {\displaystyle f(A)} is the set { f ( x ) | x ∈ A } {\displaystyle \{f(x)|x\in A\}} . This definition can also be extended to metric and topological spaces.
Relationship to continuity[edit]
The notion of the limit of a function is very closely related to the concept of continuity. A function ƒ is said to be continuous at c if it is both defined at c and its value at c equals the limit of f as x approaches c:
lim x → c f ( x ) = f ( c ) . {\displaystyle \lim _{x\to c}f(x)=f(c).}
(We have here assumed that c is a limit point of the domain of f.)
Properties[edit]
If a function f is real-valued, then the limit of f at p is L if and only if both the right-handed limit and left-handed limit of f at p exist and are equal to L.
The function f is continuous at p if and only if the limit of f(x) as x approaches p exists and is equal to f(p). If f : M → N is a function between metric spaces M and N, then it is equivalent that f transforms every sequence in M which converges towards p into a sequence in N which converges towards f(p).
If N is a normed vector space, then the limit operation is linear in the following sense: if the limit of f(x) as x approaches p is L and the limit of g(x) as x approaches p is P, then the limit of f(x) + g(x) as x approaches p is L + P. If a is a scalar from the base field, then the limit of af(x) as x approaches p is aL.
If f is a real-valued (or complex-valued) function, then taking the limit is compatible with the algebraic operations, provided the limits on the right sides of the equations below exist (the last identity only holds if the denominator is non-zero). This fact is often called the algebraic limit theorem.
lim x → p ( f ( x ) + g ( x ) ) = lim x → p f ( x ) + lim x → p g ( x ) lim x → p ( f ( x ) − g ( x ) ) = lim x → p f ( x ) − lim x → p g ( x ) lim x → p ( f ( x ) ⋅ g ( x ) ) = lim x → p f ( x ) ⋅ lim x → p g ( x ) lim x → p ( f ( x ) / g ( x ) ) = lim x → p f ( x ) / lim x → p g ( x ) {\displaystyle {\begin{matrix}\lim \limits _{x\to p}&(f(x)+g(x))&=&\lim \limits _{x\to p}f(x)+\lim \limits _{x\to p}g(x)\\\lim \limits _{x\to p}&(f(x)-g(x))&=&\lim \limits _{x\to p}f(x)-\lim \limits _{x\to p}g(x)\\\lim \limits _{x\to p}&(f(x)\cdot g(x))&=&\lim \limits _{x\to p}f(x)\cdot \lim \limits _{x\to p}g(x)\\\lim \limits _{x\to p}&(f(x)/g(x))&=&{\lim \limits _{x\to p}f(x)/\lim \limits _{x\to p}g(x)}\end{matrix}}}
In each case above, when the limits on the right do not exist, or, in the last case, when the limits in both the numerator and the denominator are zero, nonetheless the limit on the left, called an indeterminate form, may still exist—this depends on the functions f and g. These rules are also valid for one-sided limits, for the case p = ±∞, and also for infinite limits using the rules
q + ∞ = ∞ for q ≠ −∞
q × ∞ = ∞ if q > 0
q × ∞ = −∞ if q < 0
q / ∞ = 0 if q ≠ ± ∞
(see extended real number line).
Note that there is no general rule for the case q / 0; it all depends on the way 0 is approached. Indeterminate forms—for instance, 0/0, 0×∞, ∞−∞, and ∞/∞—are also not covered by these rules, but the corresponding limits can often be determined with L'Hôpital's rule or the Squeeze theorem.
Limits of compositions of functions[edit]
In general, from knowing that
lim y → b f ( y ) = c {\displaystyle \lim _{y\to b}f(y)=c} and lim x → a g ( x ) = b {\displaystyle \lim _{x\to a}g(x)=b} ,
it does not follow that lim x → a f ( g ( x ) ) = c {\displaystyle \lim _{x\to a}f(g(x))=c} . However, this "chain rule" does hold if one of the following additional conditions holds:
f(b) = c (that is, f is continuous at b), or
g does not take the value b near a (that is, there exists a δ > 0 {\displaystyle \delta >0} such that if 0 < | x − a | < δ {\displaystyle 0<|x-a|<\delta } then | g ( x ) − b | > 0 {\displaystyle |g(x)-b|>0} ).
As an example of this phenomenon, consider the following functions that violates both additional restrictions:
f ( x ) = g ( x ) = { 0 if x ≠ 0 1 if x = 0 . {\displaystyle f(x)=g(x)={\begin{cases}0&{\text{if }}x\neq 0\\1&{\text{if }}x=0\end{cases}}.}
Since the value at f(0) is a removable discontinuity,
lim x → a f ( x ) = 0 {\displaystyle \lim _{x\to a}f(x)=0} for all a {\displaystyle a} .
Thus, the naïve chain rule would suggest that the limit of f(f(x)) is 0. However, it is the case that
f ( f ( x ) ) = { 1 if x ≠ 0 0 if x = 0 {\displaystyle f(f(x))={\begin{cases}1&{\text{if }}x\neq 0\\0&{\text{if }}x=0\end{cases}}}
and so
lim x → a f ( f ( x ) ) = 1 {\displaystyle \lim _{x\to a}f(f(x))=1} for all a {\displaystyle a} .
Limits of special interest[edit]
Rational functions[edit]
For n {\displaystyle n} a nonnegative integer and constants a 1 , a 2 , a 3 , … , a n {\displaystyle a_{1},a_{2},a_{3},\ldots ,a_{n}} and b 1 , b 2 , b 3 , … , b n {\displaystyle b_{1},b_{2},b_{3},\ldots ,b_{n}} ,
lim x → ∞ a 1 x n + a 2 x n − 1 + a 3 x n − 2 + . . . + a n b 1 x n + b 2 x n − 1 + b 3 x n − 2 + . . . + b n = a 1 b 1 {\displaystyle \lim _{x\to \infty }{\frac {a_{1}{x}^{n}+a_{2}{x}^{n-1}+a_{3}{x}^{n-2}+...+a_{n}}{b_{1}{x}^{n}+b_{2}{x}^{n-1}+b_{3}{x}^{n-2}+...+b_{n}}}={\frac {a_{1}}{b_{1}}}}
This can be proven by dividing both the numerator and denominator by x n {\displaystyle x^{n}} . If the numerator is a polynomial of higher degree, the limit does not exist. If the denominator is of higher degree, the limit is 0.
Trigonometric functions[edit]
lim x → 0 sin  x x = 1 {\displaystyle \lim _{x\to 0}{\frac {\sin x}{x}}=1}
lim x → 0 1 − cos  x x = 0 {\displaystyle \lim _{x\to 0}{\frac {1-\cos x}{x}}=0}
lim x → ∞ x sin  ( 1 x ) = 1 {\displaystyle \lim _{x\to \infty }x\sin \left({\frac {1}{x}}\right)=1}
Exponential functions[edit]
lim x → 0 ( 1 + x ) 1 x = lim r → ∞ ( 1 + 1 r ) r = e {\displaystyle \lim _{x\to 0}(1+x)^{\frac {1}{x}}=\lim _{r\to \infty }\left(1+{\frac {1}{r}}\right)^{r}=e}
lim x → 0 e x − 1 x = 1 {\displaystyle \lim _{x\to 0}{\frac {e^{x}-1}{x}}=1}
lim x → 0 e a x − 1 b x = a b {\displaystyle \lim _{x\to 0}{\frac {e^{ax}-1}{bx}}={\frac {a}{b}}}
lim x → 0 c a x − 1 b x = a b ln  c {\displaystyle \lim _{x\to 0}{\frac {c^{ax}-1}{bx}}={\frac {a}{b}}\ln c}
lim x → 0 + x x = 1 {\displaystyle \lim _{x\to 0^{+}}x^{x}=1}
Logarithmic functions[edit]
lim x → 0 ln  ( 1 + x ) x = 1 {\displaystyle \lim _{x\to 0}{\frac {\ln(1+x)}{x}}=1}
lim x → 0 ln  ( 1 + a x ) b x = a b {\displaystyle \lim _{x\to 0}{\frac {\ln(1+ax)}{bx}}={\frac {a}{b}}}
lim x → 0 log c  ( 1 + a x ) b x = a b ln  c {\displaystyle \lim _{x\to 0}{\frac {\log _{c}(1+ax)}{bx}}={\frac {a}{b\ln c}}}
L'Hôpital's rule[edit]
Main article: l'Hôpital's rule
This rule uses derivatives to find limits of indeterminate forms 0/0 or ±∞/∞, and only applies to such cases. Other indeterminate forms may be manipulated into this form. Given two functions f(x) and g(x), defined over an open interval I containing the desired limit point c, then if:
lim x → c f ( x ) = lim x → c g ( x ) = 0 , {\displaystyle \lim _{x\to c}f(x)=\lim _{x\to c}g(x)=0,} or lim x → c f ( x ) = ± lim x → c g ( x ) = ± ∞ {\displaystyle \lim _{x\to c}f(x)=\pm \lim _{x\to c}g(x)=\pm \infty } , and
f {\displaystyle f} and g {\displaystyle g} are differentiable over I ∖ { c } {\displaystyle I\setminus \{c\}} , and
g ′ ( x ) ≠ 0 {\displaystyle g'(x)\neq 0} for all x ∈ I ∖ { c } {\displaystyle x\in I\setminus \{c\}} , and
lim x → c f ′ ( x ) g ′ ( x ) {\displaystyle \lim _{x\to c}{\frac {f'(x)}{g'(x)}}} exists,
then:
lim x → c f ( x ) g ( x ) = lim x → c f ′ ( x ) g ′ ( x ) {\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}}
Normally, the first condition is the most important one.
For example: lim x → 0 sin  ( 2 x ) sin  ( 3 x ) = lim x → 0 2 cos  ( 2 x ) 3 cos  ( 3 x ) = 2 ⋅ 1 3 ⋅ 1 = 2 3 . {\displaystyle \lim _{x\to 0}{\frac {\sin(2x)}{\sin(3x)}}=\lim _{x\to 0}{\frac {2\cos(2x)}{3\cos(3x)}}={\frac {2\cdot 1}{3\cdot 1}}={\frac {2}{3}}.}
Summations and integrals[edit]
Specifying an infinite bound on a summation or integral is a common shorthand for specifying a limit.
A short way to write the limit lim n → ∞ ∑ i = s n f ( i ) {\displaystyle \lim _{n\to \infty }\sum _{i=s}^{n}f(i)} is ∑ i = s ∞ f ( i ) {\displaystyle \sum _{i=s}^{\infty }f(i)} . An important example of limits of sums such as these are series.
A short way to write the limit lim x → ∞ ∫ a x f ( t ) d t {\displaystyle \lim _{x\to \infty }\int _{a}^{x}f(t)\;dt} is ∫ a ∞ f ( t ) d t {\displaystyle \int _{a}^{\infty }f(t)\;dt} .
A short way to write the limit lim x → − ∞ ∫ x b f ( t ) d t {\displaystyle \lim _{x\to -\infty }\int _{x}^{b}f(t)\;dt} is ∫ − ∞ b f ( t ) d t {\displaystyle \int _{-\infty }^{b}f(t)\;dt} .
See also[edit]
List of limits
One-sided limit
Limit of a sequence
Net (topology)
Big O notation
Limit superior and limit inferior
l'Hôpital's rule
Squeeze theorem
Non-standard calculus
Notes[edit]
Jump up ^ For example, "Limit" at Encyclopaedia of Mathematics
Jump up ^ Keisler, H. Jerome (2008), "Quantifiers in limits" (PDF), Andrzej Mostowski and foundational studies, IOS, Amsterdam, pp. 151–170
Jump up ^ Hrbacek, K. (2007), "Stratified Analysis?", in Van Den Berg, I.; Neves, V., The Strength of Nonstandard Analysis, Springer
Jump up ^ Bŀaszczyk, Piotr; Katz, Mikhail; Sherry, David (2012), "Ten misconceptions from the history of analysis and their debunking", Foundations of Science, 18 (1), arXiv:1202.4153  , doi:10.1007/s10699-012-9285-8
References[edit]
MacTutor History of Weierstrass.
MacTutor History of Bolzano
Visual Calculus by Lawrence S. Husch, University of Tennessee (2001)
Apostol, Tom M., Mathematical Analysis, 2nd ed. Addison–Wesley, 1974. ISBN 0-201-00288-4.
Bartle, Robert (1967), The elements of real analysis, Wiley
Burton, David M. (1997), The History of Mathematics: An introduction (Third ed.), New York: McGraw–Hill, pp. 558–559, ISBN 0-07-009465-9
Felscher, Walter (2000), "Bolzano, Cauchy, Epsilon, Delta", American Mathematical Monthly, Mathematical Association of America, 107 (9): 844–862, doi:10.2307/2695743, JSTOR 2695743 .
Courant, Richard (1924), Vorlesungen über Differential- und Integralrechnung, Springer Verlag
Grabiner, Judith V. (1983), "Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus", American Mathematical Monthly, Mathematical Association of America, 90 (3): 185–194, doi:10.2307/2975545, JSTOR 2975545, collected in Who Gave You the Epsilon? ISBN 978-0-88385-569-0 pp. 5–13 . Also available at: http://www.maa.org/pubs/Calc_articles/ma002.pdf
Hardy, G.H. (1921), A course in pure mathematics, Cambridge University Press
Hubbard, John H. (2015), Vector calculus, linear algebra, and differential forms: A unified approach (Fifth ed.), Matrix Editions
Miller, Jeff (1 December 2004), Earliest Uses of Symbols of Calculus, retrieved 2008-12-18 .
Page, Warren; Hersh, Reuben; Selden, Annie; et al., eds. (2002), "Media Highlights", The College Mathematics, Mathematical Association of America, 33 (2): 147–154, JSTOR Journal 2687124 Journal .
Sutherland, W. A., Introduction to Metric and Topological Spaces. Oxford University Press, Oxford, 1975. ISBN 0-19-853161-3.
Rudin, Walter (1964), Principles of mathematical analysis, McGraw-Hill
Whittaker; Watson (1904), A course of modern analysis, Cambridge University Press
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values: {\displaystyle \lim _{x\to p^{-}}f(x)=L}

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values: L={\underset  {x\in S}{\lim _{{x\to p}}}}f(x).

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values: f(x)=\begin{cases}\sin\frac{5}{x-1} & \text{ for } x< 1 \\ 0 & \text{ for } x=1 \\ \frac{0.1}{x-1}& \text{ for } x>1\end{cases}

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values: f(x)=\begin{cases}1 &  x \text{ rational } \\ 0 &  x \text{ irrational }\end{cases}

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values: f(x)=\begin{cases}x &  x \text{ rational } \\ 0 &  x \text{ irrational }\end{cases}

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values: f(x)=\begin{cases}|x| &  x \text{ rational } \\ 0 &  x \text{ irrational }\end{cases}

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values: \lim_{x \to \infty} f(x) = \infty, \lim_{x \to a^+}f(x) = -\infty. \,

values: {\displaystyle \lim _{x\to 0^{+}}\ln x=-\infty .\,}

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values: Edit section: Relationship to continuity

values: {\displaystyle \lim _{x\to c}f(x)=f(c).}

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\begin{matrix}
\lim\limits_{x \to p} & (f(x) + g(x)) & = & \lim\limits_{x \to p} f(x) + \lim\limits_{x \to p} g(x) \\
\lim\limits_{x \to p} & (f(x) - g(x)) & = & \lim\limits_{x \to p} f(x) - \lim\limits_{x \to p} g(x) \\
\lim\limits_{x \to p} & (f(x)\cdot g(x)) & = & \lim\limits_{x \to p} f(x) \cdot \lim\limits_{x \to p} g(x) \\
\lim\limits_{x \to p} & (f(x)/g(x)) & = & {\lim\limits_{x \to p} f(x) / \lim\limits_{x \to p} g(x)}
\end{matrix}

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/wiki/Removable_discontinuity
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Removable discontinuity

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values: f(f(x))=\begin{cases}1 & \text{if } x\neq 0 \\ 0 & \text{if } x=0 \end{cases}

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values: Edit section: Limits of special interest

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values: {\displaystyle \lim _{x\to 0}{\frac {\sin x}{x}}=1}

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values: {\displaystyle \lim _{x\to 0}{\frac {1-\cos x}{x}}=0}

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values: \lim_{x \to \infty} x \sin \left(\frac{1}{x}\right) = 1

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values: {\displaystyle \lim _{x\to 0}{\frac {e^{x}-1}{x}}=1}

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values: \lim_{x \to 0} \frac{c^{ax}-1}{bx} = \frac{a}{b}\ln c

values: {\displaystyle \lim _{x\to 0^{+}}x^{x}=1}

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values: {\displaystyle \lim _{x\to 0}{\frac {\ln(1+x)}{x}}=1}

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values: \lim_{x \to 0} \frac{\ln(1+ax)}{bx} = \frac{a}{b}

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values: \lim_{x \to 0} \frac{\log_c(1+ax)}{bx} = \frac{a}{b\ln c}

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/wiki/Indeterminate_forms
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Indeterminate forms

values: {\displaystyle \lim _{x\to c}f(x)=\lim _{x\to c}g(x)=0,}

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values: \lim_{x \to c}f(x)=\lim_{x \to c}g(x)=0,

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values: \lim_{x \to c}f(x)=\pm\lim_{x \to c}g(x) = \pm\infty

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values: {\displaystyle \lim _{x\to c}{\frac {f'(x)}{g'(x)}}}

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values: {\displaystyle \lim _{n\to \infty }\sum _{i=s}^{n}f(i)}

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values: {\displaystyle \sum _{i=s}^{\infty }f(i)}

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values: {\displaystyle \lim _{x\to \infty }\int _{a}^{x}f(t)\;dt}

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values: {\displaystyle \int _{a}^{\infty }f(t)\;dt}

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values: \lim_{x \to -\infty} \int_x^b f(t) \; dt

values: {\displaystyle \int _{-\infty }^{b}f(t)\;dt}

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