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Limit of a function
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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.
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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 elements of S distinct from p. 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 )} ). It also extends the notion of one-sided limits to the included endpoints of (half-)closed intervals, so the Square root function f(x)=√x can have limit 0 as x approaches 0 from above.
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 a limit point of 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) \ {a}. 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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value: f(x)=\begin{cases}|x| &  x \text{ rational } \\ 0 &  x \text{ irrational }\end{cases}

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value: \forall \varepsilon > 0 \; \exists c \; \forall x > c :\; |f(x) - L| < \varepsilon

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

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

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value: Edit section: Relationship to continuity

value: {\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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I Don't Like Mondays
From Wikipedia, the free encyclopedia
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"I Don't Like Mondays"
Single by The Boomtown Rats
from the album The Fine Art of Surfacing
B-side
"It's All the Rage"
Released
21 July 1979 (UK)
Format
7" vinyl
Recorded
Trident Studios
Length
4:19 (LP)
3:47 (single/video)
Label
Ensign (UK)
Columbia (US)
Songwriter(s)
Bob Geldof
Producer(s)
Phil Wainman
The Boomtown Rats singles chronology
"Rat Trap"
(1978) "I Don't Like Mondays"
(1979) "Diamond Smiles"
(1979)
"Rat Trap"
(1978) "I Don't Like Mondays"
(1979) "Diamond Smiles"
(1979)
"I Don't Like Mondays" is a song by Irish band The Boomtown Rats that was a number one single in the UK Singles Chart for four weeks during the summer of 1979,[1] and ranks as the sixth biggest British hit of 1979.[2] Written by Bob Geldof, it was the band's second number one single.
The full length version appeared on the group's third album, The Fine Art of Surfacing. It includes a reprise of the first verse, which was edited for the single release.
Contents
[hide]
1 History
2 Chart performance
2.1 Weekly singles charts
2.2 Year-end charts
3 Live performances
4 References
History[edit]
According to Geldof, he wrote the song after reading a telex report at Georgia State University's campus radio station, WRAS, on the shooting spree of 16-year-old Brenda Ann Spencer, who fired at children in a school playground at Grover Cleveland Elementary School in San Diego, California, US on 29 January 1979, killing two adults and injuring eight children and one police officer. Spencer showed no remorse for her crime and her full explanation for her actions was "I don't like Mondays. This livens up the day".[3] The song was first performed less than a month later.
Geldof explained how he wrote the song:
I was doing a radio interview in Atlanta with [Johnnie] Fingers and there was a telex machine beside me. I read it as it came out. Not liking Mondays as a reason for doing somebody in is a bit strange. I was thinking about it on the way back to the hotel and I just said 'Silicon chip inside her head had switched to overload'. I wrote that down. And the journalists interviewing her said, 'Tell me why?' It was such a senseless act. It was the perfect senseless act and this was the perfect senseless reason for doing it. So perhaps I wrote the perfect senseless song to illustrate it. It wasn't an attempt to exploit tragedy.[4]
Geldof had originally intended the song as a B-side, but changed his mind after the song was successful with audiences on the Rats' US tour.[4] Spencer's family tried unsuccessfully to prevent the single from being released in the United States.[4] Despite being a major hit in the United Kingdom, it only reached #73 on the US Billboard Hot 100.[5] The song was played regularly by album-oriented rock format radio stations in the United States throughout the 1980s, although radio stations in San Diego refrained from playing the track for some years in respect to local sensitivities about the shooting.
The strings were arranged by Fiachra Trench. Apart from Geldof, pianist Johnnie Fingers and drummer Simon Crowe are the only members of the group who appear on the recording.
In the UK the song reached Number One on the singles chart in July 1979, and won the Best Pop Song and Outstanding British Lyric categories at the Ivor Novello Awards.[6]
Chart performance[edit]
Weekly singles charts[edit]
Chart (1979)
Peak
position
Australia [7] 1
Canada [8] 4
Ireland 1
New Zealand 3
UK 1
U.S. Billboard Hot 100 73
U.S. Cash Box Top 100 84
Year-end charts[edit]
Chart (1979)
Position
Australia [7] 6
Canada [9] 117
New Zealand [10] 25
UK [11] 4
Chart (1980)
Position
Canada 55
Live performances[edit]
On 9 September 1981, Geldof was joined on stage by fellow Boomtown Rat, Johnnie Fingers, to perform the song for The Secret Policeman's Ball sponsored by Amnesty International. A recording of that performance appears on the 1982 album, The Secret Policeman's Other Ball.
The Boomtown Rats performed the song for Live Aid at Wembley Stadium in 1985. This was the band's final major appearance. On singing the line, "And the lesson today is how to die", Geldof paused for 20 seconds while the crowd applauded on the significance to those starving in Africa that Live Aid was intended to help.
At a concert in London in 1995, almost ten years later to the day, Bon Jovi covered the song after being joined on stage by Geldof at Wembley Stadium. This recorded performance features on Bon Jovi's live album One Wild Night Live 1985–2001, as well as on the bonus 2-CD edition of These Days. Bon Jovi was again joined by Geldof for a performance of the song at The O2 Arena on 23 June 2010, the 10th night of their 12-night residency. Geldof himself performed a version of the song while hosting the Live 8 concert in London, on 2 July 2005.[citation needed]
"I Don't Like Mondays" was subsequently covered by Tori Amos on her 2001 album Strange Little Girls and later by G4 on their 2006 album Act Three.
References[edit]
Jump up ^ Roberts, David (2006). British Hit Singles & Albums (19th ed.). London: Guinness World Records Limited. pp. 370–1. ISBN 1-904994-10-5.
Jump up ^ "Top 100 1979 - UK Music Charts". Uk-charts.top-source.info. Retrieved 2015-10-20.
Jump up ^ Mikkelson, Barbara (29 September 2005). "Urban Legends Reference Pages: Music (I Don't Like Mondays)". snopes.com.
^ Jump up to: a b c Clarke, Steve (18–31 October 1979). The Fastest Lip on Vinyl. Smash Hits. EMAP National Publications Ltd. pp. 6–7.
Jump up ^ Whitburn, Joel (2000). Top Pop Singles 1955-1999. Record Research Inc. p. 65. ISBN 0-89820-139-X.
Jump up ^ "I Don't Like Mondays". Songfacts.com. Retrieved 2009-04-05.
^ Jump up to: a b Steffen Hung. "Forum - 1970 (ARIA Charts: Special Occasion Charts)". australian-charts.com. Archived from the original on 2 June 2016. Retrieved 20 October 2015.
Jump up ^ "Image : RPM Weekly - Library and Archives Canada". Bac-lac.gc.ca. Retrieved 2015-10-20.
Jump up ^ "Image : RPM Weekly - Library and Archives Canada". Bac-lac.gc.ca. Retrieved 2015-10-20.
Jump up ^ "Top Selling Singles of 1979 | The Official New Zealand Music Chart". Nztop40.co.nz. 1979-12-31. Retrieved 2016-10-15.
Jump up ^ "Top 100 Singles of 1979" Record Mirror 5 January 1980: 30
Preceded by
"Are 'Friends' Electric?" by Tubeway Army UK number-one single
28 July 1979 - 24 August 1979 Succeeded by
"We Don't Talk Anymore" by Cliff Richard
Preceded by
"Do You Want Your Oul Lobby Washed Down" by Brendan Shine Irish Singles Chart number-one single
4 August 1979 - 31 August 1979 Succeeded by
"Viva Il Papa" by Catriona Walsh
Preceded by
"Born to Be Alive" by Patrick Hernandez Australian Kent Music Report number-one single
12 November 1979 - 25 November 1979 Succeeded by
"Video Killed the Radio Star" by The Buggles
[hide]
v
t
e
The Boomtown Rats
Bob Geldof
Pete Briquette
Gerry Cott
Simon Crowe
Johnnie Fingers
Garry Roberts
Studio albums
The Boomtown Rats
A Tonic for the Troops
The Fine Art of Surfacing
Mondo Bongo
V Deep
In the Long Grass
Singles
"Lookin' After No. 1"
"Mary of the 4th Form"
"She's So Modern"
"Like Clockwork"
"Rat Trap"
"I Don't Like Mondays"
"Diamond Smiles"
"Someone's Looking at You"
"Banana Republic"
"Up All Night"
"The Elephant's Graveyard (Guilty)"
"Never in a Million Years"
"House on Fire"
"Charmed Lives"
"Tonight"
"Drag Me Down"
"Dave"
"A Hold of Me"
Other albums
Ratrospective
The Boomtown Rats' Greatest Hits
Loudmouth – The Best of Bob Geldof and the Boomtown Rats
The Best of The Boomtown Rats
Back to Boomtown: Classic Rats Hits
So Modern – The Boomtown Rats Collection
Related articles
Discography
Ensign Records
Mercury Records
Retrieved from "https://en.wikipedia.org/w/index.php?title=I_Don%27t_Like_Mondays&oldid=813165947"
Categories:
1979 singles
European Hot 100 Singles number-one singles
Irish Singles Chart number-one singles
Songs based on actual events
Songs inspired by deaths
Songs about school
Songs about the media
Songs written by Bob Geldof
The Boomtown Rats songs
UK Singles Chart number-one singles
Song recordings produced by Phil Wainman
1979 songs
Ensign Records singles
Columbia Records singles
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Part of the constellation of Carina, Epsilon Carinae is an example of a double star featuring a main-sequence B-type star.
Typical Stellar Properties[1]
Spectral
Type
Radius
R☉
Mass
M☉
Teff
(K)
log g
B0V 10.00 17.00 25,000 4.0
B1V 6.42 13.21 25,400 3.9
B2V 5.33 9.11 20,800 3.9
B3V 4.80 7.60 18,800 4.0
B5V 3.90 5.90 15,200 4.0
B6V 3.56 5.17 13,800 4.0
B7V 3.28 4.45 12,400 4.1
B8V 3.00 3.80 11,400 4.1
B9V 2.70 3.29 10,600 4.1
A B-type main-sequence star (B V) is a main-sequence (hydrogen-burning) star of spectral type B and luminosity class V. These stars have from 2 to 16 times the mass of the Sun and surface temperatures between 10,000 and 30,000 K.[2] B-type stars are extremely luminous and blue. Their spectra have neutral helium, which are most prominent at the B2 subclass, and moderate hydrogen lines. Examples include Regulus and Algol A.[3]
This class of stars was introduced with the Harvard sequence of stellar spectra and published in the Revised Harvard photometry catalogue. The definition of type B-type stars was the presence of non-ionized helium lines with the absence of singly ionized helium in the blue-violet portion of the spectrum. All of the spectral classes, including the B type, were subdivided with a numerical suffix that indicated the degree to which they approached the next classification. Thus B2 is two tenths of the way from type B (or B0) to type A.[4][5]
Later, however, more refined spectra showed lines of ionized helium for stars of type B0. Likewise, A0 stars also show weak lines of non-ionized helium. Subsequent catalogues of stellar spectra classified the stars based on the strengths of absorption lines at specific frequencies, or by comparing the strengths of different lines. Thus, in the MK Classification system, the spectral class B0 has the line at wavelength 438.7 nm being stronger than the line at 420.0 nm.[6] The Balmer series of hydrogen lines grows stronger through the B class, then peak at type A2. The lines of ionized silicon are used to determine the sub-class of the B-type stars, while magnesium lines are used to distinguish between the temperature classes.[5]
Type-B stars do not have a corona and lack a convection zone in their outer atmosphere. They have a higher mass loss rate than smaller stars such as the Sun, and their stellar wind has velocities of about 3,000 km/s.[7] The energy generation in main-sequence B-type stars comes from the CNO cycle of thermonuclear fusion. Because the CNO cycle is very temperature sensitive, the energy generation is heavily concentrated at the center of the star, which results in a convection zone about the core. This results in a steady mixing of the hydrogen fuel with the helium byproduct of the nuclear fusion.[8] Many B-type stars have a rapid rate of rotation, with an equatorial rotation velocity of about 200 km/s.[9]
Contents
[hide]
1 Be and B(e) stars
2 Spectral Standard Stars
3 Chemical peculiarities
4 Planets
5 See also
6 References
Be and B(e) stars[edit]
Main article: B(e) stars
Spectral objects known as "Be stars" are massive yet non-supergiant entities that notably have, or had at some time, one or more Balmer lines in emission, with the hydrogen-related electromagnetic radiation series projected out by the stars being of particular scientific interest. Be stars are generally thought to feature unusually strong stellar winds, high surface temperatures, and significant attrition of stellar mass as the objects rotate at a curiously rapid rate, all of this in contrast to many other main-sequence star types.[10]
Though the related terminologies are confusingly ambiguous, spectral objects known as "B(e)" or "B[e] stars" are distinct from Be stars since said B(e) entities are in possession of distinctive neutral or low ionization emission lines that are considered to have 'forbidden mechanisms', something denoted by the use of brackets or parenthesis. In other words, these particular stars' emissions appear to undergo processes not normally allowed under first-order perturbation theory in quantum mechanics. It should be noted that the definition of a "B(e) star" can include objects that are large enough to be in Blue giant and Blue supergiant territory, beyond the size of standard main-sequence stars.
Spectral Standard Stars[edit]
The revised Yerkes Atlas system (Johnson & Morgan 1953)[11] listed a dense grid of B-type dwarf spectral standard stars, however not all of these have survived to this day as standards. The "anchor points" of the MK spectral classification system among the B-type main-sequence dwarf stars, i.e. those standard stars that have remain unchanged since at least the 1940s, are upsilon Orionis (B0 V), eta Aurigae (B3 V), eta Ursae Majoris (B3 V).[12][13] Besides these anchor standards, the seminal review of MK classification by Morgan & Keenan (1973)[13] listed "dagger standards" of Tau Scorpii (B0 V), Omega Scorpii (B1 V), 42 Orionis (B1 V), 22 Scorpii (B2 V), Rho Aurigae (B5 V), and 18 Tau (B8 V). The Revised MK Spectra Atlas of Morgan, Abt, & Tapscott (1978)[14] further contributed the standards beta2 Sco (B2 V), 29 Persei (B3 V), HD 36936 (B5 V), and HD 21071 (B7 V). Gray & Garrison (1994)[15] contributed two B9 V standards: omega For A and HR 2328. The only published B4 V standard is 90 Leonis, from Lesh (1968).[16] There has been little agreement in the literature on choice of B6 V standard.
Chemical peculiarities[edit]
Some of the B-type stars of stellar class B0–B3 exhibit unusually strong lines of non-ionized helium. These chemically peculiar stars are termed helium-strong stars. These often have strong magnetic fields in their photosphere. In contrast, there are also helium-weak B-type stars with understrength helium lines and strong hydrogen spectra. Other chemically peculiar B-types stars are mercury–manganese stars with spectral types B7-B9. Finally, the aforementioned Be stars show a prominent emission spectrum of hydrogen.[17]
Planets[edit]
B-type stars known to have planets include the main-sequence B-types HIP 78530 b, the subgiants Kappa Andromedae b and a few B-type subdwarfs.
See also[edit]
Herbig Ae/Be star
Stellar classification, Class B
Star count
References[edit]
Jump up ^ Silaj, J.; et al. (November 2014), "The Hα Profiles of Be Shell Stars", The Astrophysical Journal, 795 (1): 12, Bibcode:2014ApJ...795...82S, doi:10.1088/0004-637X/795/1/82, 82.
Jump up ^ Habets, G. M. H. J.; Heintze, J. R. W. (November 1981). "Empirical bolometric corrections for the main-sequence". Astronomy and Astrophysics Supplement. 46: 193–237. Bibcode:1981A&AS...46..193H. , Tables VII and VIII.
Jump up ^ SIMBAD, entries on Regulus and Algol A, accessed June 19, 2007.
Jump up ^ Pickering, Edward Charles (1908). "Revised Harvard photometry : a catalogue of the positions, photometric magnitudes and spectra of 9110 stars, mainly of the magnitude 6.50, and brighter observed with the 2 and 4 inch meridian photometers". Annals of the Astronomical Observatory of Harvard College. 50. Bibcode:1908AnHar..50....1P. Retrieved 2009-09-21.
^ Jump up to: a b Gray, C. Richard O.; Corbally, J. (2009). Stellar Spectral Classification. Princeton University Press. pp. 115–122. ISBN 0691125112.
Jump up ^ Morgan, William Wilson; Keenan, Philip Childs; Kellman, Edith (1943). An atlas of stellar spectra, with an outline of spectral classification. Chicago, Ill: The University of Chicago press. Bibcode:1943assw.book.....M.
Jump up ^ Aschenbach, B.; Hahn, Hermann-Michael; Truemper, Joachim (1998). Hermann-Michael Hahn, ed. The invisible sky: ROSAT and the age of X-ray astronomy. Springer. p. 76. ISBN 0387949283.
Jump up ^ Böhm-Vitense, Erika (1992). Introduction to stellar astrophysics. 3. Cambridge University Press. p. 167. ISBN 0521348714.
Jump up ^ McNally, D. (1965). "The distribution of angular momentum among main sequence stars". The Observatory. 85: 166–169. Bibcode:1965Obs....85..166M.
Jump up ^ Slettebak, Arne (July 1988). "The Be Stars". Publications of the Astronomical Society of the Pacific. 100: 770–784. Bibcode:1988PASP..100..770S. doi:10.1086/132234.
Jump up ^ Fundamental stellar photometry for standards of spectral type on the revised system of the Yerkes spectral atlas H.L. Johnson & W.W. Morgan, 1953, Astrophysical Journal, 117, 313
Jump up ^ MK ANCHOR POINTS, Robert F. Garrison
^ Jump up to: a b Spectral Classification, W.W. Morgan & P.C. Keenan, 1973, Annual Review of Astronomy and Astrophysics, vol. 11, p.29
Jump up ^ Revised MK Spectral Atlas for stars earlier than the sun, W.W. Morgan, W. W., H.A. Abt, J.W. Tapscott, 1978, Williams Bay: Yerkes Observatory, and Tucson: Kitt Peak National Observatory
Jump up ^ The late B-type stars: Refined MK classification, confrontation with stromgren photometry, and the effects of rotation, R.F. Gray & R.O. Garrison, 1994, The Astronomical Journal, vol. 107, no. 4, p. 1556-1564
Jump up ^ The Kinematics of the Gould Belt: an Expanding Group? J.R. Lesh, 1968, Astrophysical Journal Supplement, vol. 17, p.371 (Table 1)
Jump up ^ Gray, Richard O.; Corbally, C. J. (2009). Stellar Spectral Classification. Princeton University Press. pp. 123–136. ISBN 0691125112.
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