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[edit]
[edit]
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[edit]
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
Operations on two known limits[edit]
If and then:
Limits involving derivatives or infinitesimal changes[edit]
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[edit]
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,
Polynomials and functions of the form xa[edit]
Polynomials in x[edit]
In general, if is a polynomial then, by the continuity of polynomials,[5]
Functions of the form xa[edit]
Exponential functions[edit]
Functions of the form ag(x)[edit]
- , due to the continuity of
- [6]
Functions of the form xg(x)[edit]
Functions of the form f(x)g(x)[edit]
- [2]
- [2]
- [7]
- [6]
- . This limit can be derived from this limit.
Sums, products and composites[edit]
Logarithmic functions[edit]
Natural logarithms[edit]
- , due to the continuity of . In particular,
- [7]
- . This limit follows from L'Hôpital's rule.
- , hence
- [6]
Logarithms to arbitrary bases[edit]
For b > 1,
For b < 1,
Both cases can be generalized to:
where and is the Heaviside step function
Trigonometric functions[edit]
If is expressed in radians:
These limits both follow from the continuity of sin and cos.
- .[7][8] Or, in general,
- , for a not equal to 0.
- , for b not equal to 0.
- [4][8][9]
- , 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[edit]
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[edit]
- . This can be proven by considering the inequality at .
- . This can be derived from Viète's formula for π.
Limiting behavior[edit]
Asymptotic equivalences[edit]
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[edit]
The behaviour of functions described by Big O notation can also be described by limits. For example
- if
References[edit]
- ^ a b c d e f g h i j "Basic Limit Laws". math.oregonstate.edu. Retrieved 2019-07-31.
- ^ a b c d e f g h i j k l "Limits Cheat Sheet - Symbolab". www.symbolab.com. Retrieved 2019-07-31.
- ^ a b c d e f g h "Section 2.3: Calculating Limits using the Limit Laws" (PDF).
- ^ a b c "Limits and Derivatives Formulas" (PDF).
- ^ a b c d e f "Limits Theorems". archives.math.utk.edu. Retrieved 2019-07-31.
- ^ a b c d e "Some Special Limits". www.sosmath.com. Retrieved 2019-07-31.
- ^ a b c d "SOME IMPORTANT LIMITS - Math Formulas - Mathematics Formulas - Basic Math Formulas". www.pioneermathematics.com. Retrieved 2019-07-31.
- ^ a b "World Web Math: Useful Trig Limits". Massachusetts Institute of Technology. Retrieved 2023-03-20.
- ^ "Calculus I - Proof of Trig Limits". Retrieved 2023-03-20.