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This is a list of limits for common functions such as elementary functions. In this article, the terms a, b and c are constants with respect to x.
Limits for general functions[]
Definitions of limits and related concepts[]
if and only if . This is the (ε, δ)-definition of limit.
The limit superior and limit inferior 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:
- [1][2][3]
- [4] if L is not equal to 0.
- if n is a positive integer[1][2][3]
- if n is a positive integer, and if n is even, then L > 0.[1][3]
In general, if g(x) is continuous at L and then
- [1][2]
Operations on two known limits[]
If and then:
- [1][2][3]
- [1][2][3]
- [1][2][3]
Limits involving derivatives or infinitesimal changes[]
In these limits, the infinitesimal change is often denoted or . If is differentiable at ,
- . This is the definition of the derivative. 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.
- . This is the product rule.
If and are differentiable on an open interval containing c, except possibly c itself, and , L'Hôpital's rule can be used:
- [2]
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[5]
If and for all x in an open interval that contains c, except possibly c itself,
This is known as the
squeeze theorem.
[1][2] 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[]
- [1][2][3]
Polynomials in x[]
- [1][2][3]
- if n is a positive integer[5]
In general, if is a polynomial then, by the continuity of polynomials,[5]
This is also true for
rational functions, as they are continuous on their
domains.
[5]
Functions of the form xa[]
- [5] In particular,
- .[5] In particular,
- [6]
Exponential functions[]
Functions of the form ag(x)[]
- , due to the continuity of
- [6]
Functions of the form xg(x)[]
Functions of the form f(x)g(x)[]
- [2]
- [2]
- [7]
- [6]
- . This limit can be derived from this limit.
Sums, products and composites[]
- for all positive a.[4][7]
Logarithmic functions[]
Natural logarithms[]
- , due to the continuity of . In particular,
- [7]
- . This limit follows from L'Hôpital's rule.
- [6]
Logarithms to arbitrary bases[]
For b > 1,
For b < 1,
Both cases can be generalized to:
where and is the Heaviside step function
Trigonometric functions[]
If is expressed in radians:
These limits both follow from the continuity of sin and cos.
- .[7] Or, in general,
- , for a not equal to 0.
- , for b not equal to 0.
- [4]
- , for integer n.
- , where x0 is an arbitrary real number.
- , where d is Dottie number. x0 can be any arbitrary real number.
Sums[]
In general, any infinite series is the limit of its partial sums. For example, an analytic function is the limit of its Taylor series, within its radius of convergence.
- . This is known as the harmonic series.[6]
- . 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 for π.
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, , where π(x) is the prime counting function.
- , due to Stirling's approximation, .
Big O notation[]
The behaviour of functions described by Big O notation can also be described by limits. For example
- if
References[]