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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
![{\displaystyle \lim _{x\to c}f(x)=f(c).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c54be0a51cc077cf4acebdeb7a40674d6b60b32)
Operations on a single known limit
If
then:
![{\displaystyle \lim _{x\to c}\,[f(x)\pm a]=L\pm a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e1d3ef86a2d1c54fd7e6cfe84ecc0c4a8809cd8)
[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.
![{\displaystyle \lim _{h\to 0}\left({\frac {f(x+h)}{f(x)}}\right)^{1/h}=\exp \left({\frac {f'(x)}{f(x)}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5cda4e61090fa1d3d709034f80fbd6104fb93729)
![{\displaystyle \lim _{h\to 0}{\left({f(e^{h}x) \over {f(x)}}\right)^{1/h}}=\exp \left({\frac {xf'(x)}{f(x)}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f8e251b5e59a5e283d2d2ef9256d703618406d9)
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]
![{\displaystyle \lim _{x\to c}f(x)\leq \lim _{x\to c}g(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a74846b4f6bf26e78591c2904629cdafed48439)
If
and
for all x in an open interval that contains c, except possibly c itself,
![{\displaystyle \lim _{x\to c}g(x)=L.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07081878c15c4b1ad07d2a3fca2d8bd0f9c63e25)
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] ![{\displaystyle \lim _{x\to c}(ax+b)=ac+b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2541766256fd7760536e6c72536a5da1c5914866)
if n is a positive integer[5] ![{\displaystyle \lim _{x\to \infty }x/a={\begin{cases}\infty ,&a>0\\{\text{does not exist}},&a=0\\-\infty ,&a<0\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d60cd118eb5756d5c47f7dd92b18e7605ac56d6b)
In general, if
is a polynomial then, by the continuity of polynomials,[5]
![{\displaystyle \lim _{x\to c}p(x)=p(c)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70f1f81905b30acc2487b9bb1f69acb26bc84d48)
This is also true for rational functions, as they are continuous on their
domains.
[5] Functions of the form xa
[5] In particular, ![{\displaystyle \lim _{x\to \infty }x^{a}={\begin{cases}\infty ,&a>0\\1,&a=0\\0,&a<0\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2afa71c8dd73f6b8bdd8f9d8bb7ddceee0f32123)
.[5] In particular,
[6]
![{\displaystyle \lim _{x\to 0^{+}}x^{-n}=\lim _{x\to 0^{+}}{\frac {1}{x^{n}}}=+\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4dc30e313a8d34e2b63817e1e40a95a409c1b7b)
![{\displaystyle \lim _{x\to 0^{-}}x^{-n}=\lim _{x\to 0^{-}}{\frac {1}{x^{n}}}={\begin{cases}-\infty ,&{\text{if }}n{\text{ is odd}}\\+\infty ,&{\text{if }}n{\text{ is even}}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14c7fdb3e8f5d090ce064183a921e59b1524d822)
![{\displaystyle \lim _{x\to \infty }ax^{-1}=\lim _{x\to \infty }a/x=0{\text{ for any real }}a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54f5f183bd64f582a6b468484c63ed0666d5566f)
Exponential functions
Functions of the form ag(x)
, due to the continuity of ![{\displaystyle e^{x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/841c0d168e64191c45a45e54c7e447defd17ec6a)
![{\displaystyle \lim _{x\to \infty }a^{x}={\begin{cases}\infty ,&a>1\\1,&a=1\\0,&0<a<1\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/264d573fbe20d8772ceb811e2f45f962daebe41b)
[6] ![{\displaystyle \lim _{x\to \infty }{\sqrt[{x}]{a}}=\lim _{x\to \infty }{a}^{1/x}={\begin{cases}1,&a>0\\0,&a=0\\{\text{does not exist}},&a<0\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/257587de98c10b53ea0f58512ac1955779cf57d1)
Functions of the form xg(x)
![{\displaystyle \lim _{x\to \infty }{\sqrt[{x}]{x}}=\lim _{x\to \infty }{x}^{1/x}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa1a821ef69483eecca48841d2b536f92682b8d6)
Functions of the form f(x)g(x)
[2]
[2] ![{\displaystyle \lim _{x\to 0}\left(1+kx\right)^{\frac {m}{x}}=e^{mk}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49cff17ec90d1b513145446d46d627a57d7a9b8f)
[7] ![{\displaystyle \lim _{x\to +\infty }\left(1-{\frac {1}{x}}\right)^{x}={\frac {1}{e}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9db2ba69ebe50f31110495ba9c7d0c45e2754281)
[6]
. This limit can be derived from this limit.
Sums, products and composites
![{\displaystyle \lim _{x\to 0}xe^{-x}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90a973a5e219efe6468257eff9270f8b2aaae5d1)
![{\displaystyle \lim _{x\to \infty }xe^{-x}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9117d6dce83cb5b774801c80fb4d4234bc8bba5d)
for all positive a.[4][7] ![{\displaystyle \lim _{x\to 0}\left({\frac {e^{x}-1}{x}}\right)=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e09d354f5b794124f39d125f679dfcdcbf8167a)
![{\displaystyle \lim _{x\to 0}\left({\frac {e^{ax}-1}{x}}\right)=a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08f2a74ac98d186f161c1f274a6bf2181dc756b0)
Logarithmic functions
Natural logarithms
, due to the continuity of
. In particular, ![{\displaystyle \lim _{x\to 0^{+}}\log x=-\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb7a454bbcf3fafcf6ea82fdcd1b8346c5c0d1a7)
![{\displaystyle \lim _{x\to \infty }\log x=\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/40a3dc39adf241c3465eb63b5e002296b7d0c57e)
![{\displaystyle \lim _{x\to 1}{\frac {\ln(x)}{x-1}}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31e8144e6486167e696a70b49e1ef2ff4019cbec)
[7]
. This limit follows from L'Hôpital's rule.
, hence ![{\displaystyle \lim _{x\to 0}x^{x}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8184fc4047cab37d6916c4aa9e346f987b2339a)
[6]
Logarithms to arbitrary bases
For b > 1,
![{\displaystyle \lim _{x\to 0^{+}}\log _{b}x=-\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bbcfd6631c47b559f3480df2b1fb842b2811ef1)
![{\displaystyle \lim _{x\to \infty }\log _{b}x=\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/88aeb1009154d3f576e56442506494c7cf2dfbe6)
For b < 1,
![{\displaystyle \lim _{x\to 0^{+}}\log _{b}x=\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa7d1ff3c33f0ce69a551b391b93390621833033)
![{\displaystyle \lim _{x\to \infty }\log _{b}x=-\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/446f76a935919673802538bb2d34d73b8f701bf1)
Both cases can be generalized to:
![{\displaystyle \lim _{x\to 0^{+}}\log _{b}x=-F(b)\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc66e3b303e99eb2ffc1b2b6f67636a823c6773e)
![{\displaystyle \lim _{x\to \infty }\log _{b}x=F(b)\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/308e56c53b99a00588342fa4b03a34efd2dfcf62)
where
and
is the Heaviside step function
Trigonometric functions
If
is expressed in radians:
![{\displaystyle \lim _{x\to a}\sin x=\sin a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04bf4a31eeee313433f14413d9e3c441d69576a9)
![{\displaystyle \lim _{x\to a}\cos x=\cos a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/233fccfb3bbfdad201662ecc5dce951fd4baf7b9)
These limits both follow from the continuity of sin and cos.
.[7][8] Or, in general,
, for a not equal to 0. ![{\displaystyle \lim _{x\to 0}{\frac {\sin ax}{x}}=a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/415364a0432ca572cee3f3ec1fa393bc925f9124)
, for b not equal to 0.
![{\displaystyle \lim _{x\to \infty }x\sin \left({\frac {1}{x}}\right)=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f11a7e97877900ffcb35de9f9636ada056329dc)
[4][8][9] ![{\displaystyle \lim _{x\to 0}{\frac {1-\cos x}{x^{2}}}={\frac {1}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/580636aba444eda89b876cfc9e36e1f31d40e771)
, 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.[6]
. This is the Euler Mascheroni constant.
Notable special limits
![{\displaystyle \lim _{n\to \infty }{\frac {n}{\sqrt[{n}]{n!}}}=e}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e67d9f7e2588c9b3d418f1107e9ea27b8f330ed)
. 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 ![{\displaystyle \limsup _{x\to \infty }{\frac {|f(x)|}{g(x)}}<\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b86030adcec1e638720c8baa963d43f3304e24d2)
References
- ^ 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.