This is a list of
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
s for common
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
s such as
elementary function
In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponen ...
s. In this article, the terms ''a'', ''b'' and ''c'' are constants with respect to SM
Limits for general functions
Definitions of limits and related concepts
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
This is the
(ε, δ)-definition of limit.
The
limit superior and limit inferior
In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a ...
of a sequence are defined as
and
.
A function,
, is said to be continuous at a point, ''c'', if
Operations on a single known limit
If
then:
*
*
*
if L is not equal to 0.
*
if ''n'' is a positive integer
*
if ''n'' is a positive integer, and if ''n'' is even, then ''L'' > 0.
In general, if ''g''(''x'') is continuous at ''L'' and
then
*
Operations on two known limits
If
and
then:
*
*
*
Limits involving derivatives or infinitesimal changes
In these limits, the infinitesimal change
is often denoted
or
. If
is
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
at
,
*
. This is the definition of the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
. All
differentiation rules can also be reframed as rules involving limits. For example, if ''g''(''x'') is differentiable at ''x'',
**
. This is the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
.
**
. This is the
product rule
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + ...
.
*
*
If
and
are differentiable on an open interval containing ''c'', except possibly ''c'' itself, and
,
L'Hôpital's rule
In calculus, l'Hôpital's rule or l'Hospital's rule (, , ), also known as Bernoulli's rule, is a theorem which provides a technique to evaluate limits of indeterminate forms. Application (or repeated application) of the rule often converts an i ...
can be used:
*
Inequalities
If
for all x in an interval that contains ''c'', except possibly ''c'' itself, and the limit of
and
both exist at ''c'', then
If
and
for all ''x'' in an
open interval
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
that contains ''c'', except possibly ''c'' itself,
This is known as the
squeeze theorem
In calculus, the squeeze theorem (also known as the sandwich theorem, among other names) is a theorem regarding the limit of a function that is trapped between two other functions.
The squeeze theorem is used in calculus and mathematical anal ...
.
This applies even in the cases that ''f''(''x'') and ''g''(''x'') take on different values at ''c'', or are discontinuous at ''c''.
Polynomials and functions of the form ''xa''
*
Polynomials in x
*
*
*
if ''n'' is a positive integer
*
In general, if
is a polynomial then, by the continuity of polynomials,
This is also true for
rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
s, as they are continuous on their
domains.
Functions of the form ''xa''
*
In particular,
**
*
.
In particular,
**
*
*
*
Exponential functions
Functions of the form ''a''''g''(''x'')
*
, due to the continuity of
*
*
*
Functions of the form ''x''''g''(''x'')
*
Functions of the form ''f''(''x'')''g''(''x'')
*
*
*
*
*
*
*
. This limit can be derived from
this limit.
Sums, products and composites
*
*
*
for all positive ''a''.
*
*
Logarithmic functions
Natural logarithms
*
, due to the continuity of
. In particular,
**
**
*
*
*
. This limit follows from
L'Hôpital's rule
In calculus, l'Hôpital's rule or l'Hospital's rule (, , ), also known as Bernoulli's rule, is a theorem which provides a technique to evaluate limits of indeterminate forms. Application (or repeated application) of the rule often converts an i ...
.
*
*
Logarithms to arbitrary bases
For ''b'' > 1,
*
*
For ''b'' < 1,
*
*
Both cases can be generalized to:
*
*
where
and
is the
Heaviside step function
The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argume ...
Trigonometric functions
If
is expressed in radians:
*
*
These limits both follow from the continuity of sin and cos.
*
.
Or, in general,
**
, for ''a'' not equal to 0.
**
**
, for ''b'' not equal to 0.
*
*
*
*
, for integer ''n''.
*
, where ''x''
0 is an arbitrary real number.
*
, where d is the
Dottie number
The Dottie number is the unique real fixed point of the cosine function.
In mathematics, the Dottie number is a constant that is the unique real root of the equation
: \cos x = x ,
where the argument of \cos is in radians. The decimal expan ...
. ''x''
0 can be any arbitrary real number.
Sums
In general, any
infinite series
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
is the limit of its
partial sums
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
. For example, an
analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex an ...
is the limit of its
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
, within its
radius of convergence
In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series co ...
.
*
. This is known as the
harmonic series.
*
. This is the
Euler Mascheroni constant.
Notable special limits
*
*
. This can be proven by considering the inequality
at
.
*
. This can be derived from
Viète's formula
In mathematics, Viète's formula is the following infinite product of nested radicals representing twice the reciprocal of the mathematical constant :
\frac2\pi = \frac2 \cdot \frac2 \cdot \frac2 \cdots
It can also be represented as:
\frac2\pi ...
for
.
Limiting behavior
Asymptotic equivalences
Asymptotic equivalences,
, are true if
. Therefore, they can also be reframed as limits. Some notable asymptotic equivalences include
*
, due to the
prime number theorem
In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying ...
,
, where π(x) is the
prime counting function.
*
, due to
Stirling's approximation
In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related but less p ...
,
.
Big O notation
The behaviour of functions described by
Big O notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Lan ...
can also be described by limits. For example
*
if
References
{{Calculus topics
Limits (mathematics)
Limits
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
Functions and mappings