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:
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 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:
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 that contains c, except possibly c itself, This is known as the squeeze theorem. 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 functions, as they are continuous on their domains.
Functions of the form xa
In particular,
. In particular,
Exponential functions
Functions of the form ag(x) , due to the continuity of
Functions of the form xg(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. , hence
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.
. Or, in general, , for a not equal to 0.
, for b not equal to 0.
, for integer n. . Or, in general, , for a not equal to 0. , for b not equal to 0. , where x0 is an arbitrary real number. , where d is the 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. . 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 .
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