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((Short description|none))
((more citations|date=August 2019))
((Expand list|date=February 2011))
This is a '''list of [[limit (mathematics)|limit]]s''' for common [[function (mathematics)|function]]s such as [[elementary function]]s. 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 ===
<math>\lim_{x \to c} f(x) = L </math> [[if and only if]] ((nowrap|<math>\forall \varepsilon>0\ \exists \delta>0 : 0 < |x-c| < \delta \implies |f(x)-L| < \varepsilon</math>.)) This is the [[(ε, δ)-definition of limit]].

The [[limit superior and limit inferior]] of a sequence are defined as <math>\limsup_{n \to \infty} x_n = \lim_{n \to \infty} \left(\sup_{m \geq n} x_m\right) </math> and <math>\liminf_{n \to \infty} x_n = \lim_{n \to \infty}\left(\inf_{m \geq n} x_m\right) </math>.

A function, <math>f(x)</math>, is said to be continuous at a point, ''c'', if <math display="block">\lim_{x \to c} f(x) = f(c).</math>

=== Operations on a single known limit ===

If <math> \lim_{x \to c} f(x) = L </math> then:

*<math>\lim_{x \to c} \, [f(x) \pm a] = L \pm a</math>
*<math>\lim_{x \to c} \, a f(x) = a L</math><ref name=":0">((Cite web|url=https://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/SandS/lHopital/limit_laws.html|title=Basic Limit Laws|website=math.oregonstate.edu|access-date=2019-07-31))</ref><ref name=":1">((Cite web|url=https://www.symbolab.com/cheat-sheets/Limits#|title=Limits Cheat Sheet - Symbolab|website=www.symbolab.com|language=en|access-date=2019-07-31))</ref><ref name=":5">((Cite web|url=http://faculty.up.edu/wootton/Calc1/Section2.3.pdf|title=Section 2.3: Calculating Limits using the Limit Laws))</ref>
*<math>\lim_{x \to c} \frac{1}{f(x)}= \frac1L</math><ref name=":6">((Cite web|url=https://www.mathportal.org/formulas/limits_and_derivatives_formulas.pdf|title=Limits and Derivatives Formulas))</ref> if L is not equal to 0.
*<math>\lim_{x \to c} \, f(x)^n = L^n </math>  if ''n''  is a positive integer<ref name=":0" /><ref name=":1" /><ref name=":5" />
*<math>\lim_{x \to c} \, f(x)^{1 \over n} = L^{1 \over n} </math> if ''n'' is a positive integer, and if ''n'' is even, then ''L'' > 0.<ref name=":0" /><ref name=":5" />
In general, if ''g''(''x'') is continuous at ''L'' and <math>\lim_{x \to c} f(x) = L</math> then
*<math>\lim_{x \to c} g\left(f(x)\right) =g(L)</math><ref name=":0" /><ref name=":1" />

=== Operations on two known limits ===
If <math> \lim_{x \to c} f(x) = L_1</math> and <math>\lim_{x \to c} g(x) = L_2</math> then:
*<math>\lim_{x \to c} \, [f(x) \pm g(x)] = L_1 \pm L_2</math><ref name=":0" /><ref name=":1" /><ref name=":5" />
*<math>\lim_{x \to c} \, [f(x)g(x)] = L_1 \cdot L_2</math><ref name=":0" /><ref name=":1" /><ref name=":5" />
*<math>\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{L_1}{L_2} \qquad \text{ if } L_2 \ne 0</math><ref name=":0" /><ref name=":1" /><ref name=":5" />

===Limits involving derivatives or infinitesimal changes===
In these limits, the infinitesimal change <math>h</math> is often denoted <math>\Delta x</math> or <math>\delta x</math>. If <math>f(x)</math>is [[differentiable function|differentiable]] at <math>x</math>, 

*<math>\lim_{h \to 0} {f(x+h)-f(x)\over h} = f'(x)</math>. 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'',
**<math>\lim_{h \to 0} {f\circ g(x+h)-f\circ g(x)\over h}=f'[g(x)]g'(x)</math>. This is the [[chain rule]].
**<math>\lim_{h \to 0} {f(x+h)g(x+h)-f(x)g(x)\over h}=f'(x)g(x)+f(x)g'(x)</math>. This is the [[product rule]].
*<math>\lim_{h \to 0} \left(\frac{f(x+h)}{f(x)}\right)^{1/h} = \exp\left(\frac{f'(x)}{f(x)}\right)</math>
*<math>\lim_{h \to 0} {\left({f(e^h x)\over{f(x)))\right)^{1/h} } = \exp\left(\frac{x f'(x)}{f(x)}\right)</math>

If <math>f(x)</math> and <math>g(x)</math> are differentiable on an open interval containing ''c'', except possibly ''c'' itself, and <math>\lim_{x \to c} f(x) = \lim_{x \to c} g(x) = 0 \text{ or } \pm\infty</math>, [[L'Hôpital's rule]] can be used:
*<math>\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}</math><ref name=":1" />

=== Inequalities ===
If <math>f(x)\leq g(x) </math> for all x in an interval that contains ''c'', except possibly ''c'' itself, and the limit of <math>f(x) </math> and <math>g(x) </math> both exist at ''c'', then<ref name=":4">((Cite web|url=http://archives.math.utk.edu/visual.calculus/1/limits.18/index.html|title=Limits Theorems|website=archives.math.utk.edu|access-date=2019-07-31))</ref>
<math display="block">\lim_{x\to c}f(x)\leq \lim_{x\to c}g(x) </math>

If <math>\lim_{x \to c} f(x) = \lim_{x \to c} h(x) = L</math> and <math>f(x)\leq g(x)\leq h(x)</math>for all ''x'' in an [[open interval]] that contains ''c'', except possibly ''c'' itself,
<math display="block">\lim_{x \to c} g(x) = L.</math> This is known as the [[squeeze theorem]].<ref name=":0" /><ref name=":1" /> 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 ''x<sup>a</sup>''==

*<math>\lim_{x \to c} a = a</math><ref name=":0" /><ref name=":1" /><ref name=":5" />

=== Polynomials in x ===

*<math>\lim_{x \to c} x = c</math><ref name=":0" /><ref name=":1" /><ref name=":5" />
*<math>\lim_{x \to c} (ax + b) = ac + b</math>
*<math>\lim_{x \to c} x^n = c^n</math> if ''n'' is a positive integer<ref name=":4" />
*<math>\lim_{x\to\infty} x/a = \begin{cases}
\infty, & a > 0 \\
\text{does not exist}, & a = 0 \\
-\infty, & a < 0 \end{cases}</math>

In general, if <math>p(x)</math>is a polynomial then, by the continuity of polynomials,<ref name=":4" /> <math display="block">\lim_{x \to c} p(x) = p(c)</math> This is also true for [[rational function]]s, as they are continuous on their [[Domain of a function|domains]].<ref name=":4" />

=== Functions of the form ''x<sup>a</sup>'' ===

*<math>\lim_{x\to c}x^a=c^a.</math><ref name=":4" /> In particular,
**<math>\lim_{x\to\infty}x^a=\begin{cases} \infty, & a > 0 \\ 1, & a = 0 \\ 0, & a < 0 \end{cases}</math>
*<math>\lim_{x\to c}x^{1/a}=c^{1/a} </math>.<ref name=":4" /> In particular,
**<math>\lim_{x\to\infty} x^{1/a}=\lim_{x\to\infty}\sqrt[a]{x}= \infty \text{ for any } a > 0 </math><ref name=":3">((Cite web| url=http://www.sosmath.com/calculus/sequence/specialim/specialim.html|title=Some Special Limits|website=www.sosmath.com|access-date=2019-07-31))</ref>
*<math>\lim_{x \to 0^+} x^{-n} =\lim_{x \to 0^+} \frac{1}{x^n}= +\infty</math>
*<math>\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} </math>
*<math>\lim_{x\to\infty} ax^{-1}=\lim_{x\to\infty}a/x=0 \text{ for any real }a </math>

==Exponential functions==

=== Functions of the form ''a''<sup>''g''(''x'')</sup> ===
*<math>\lim_{x \to c} e^{x} = e^c </math>, due to the continuity of <math>e^{x} </math>
*<math>\lim_{x\to\infty}a^x=\begin{cases} \infty, & a > 1 \\ 1, & a = 1 \\ 0, & 0 < a < 1 \end{cases}</math>
*<math>\lim_{x\to\infty}a^{-x}=\begin{cases} 0, & a > 1 \\ 1, & a = 1 \\ \infty, & 0 < a < 1 \end{cases}</math><ref name=":3" />
*<math>\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}</math>

=== Functions of the form ''x''<sup>''g''(''x'')</sup> ===
*<math>\lim_{x\to\infty}\sqrt[x]{x}=\lim_{x\to\infty}{x}^{1/x}=1</math>

=== Functions of the form ''f''(''x'')<sup>''g''(''x'')</sup> ===
*<math>\lim_{x\to+\infty} \left( \frac{x}{x+k}\right)^x=e^{-k}</math><ref name=":1" />
*<math>\lim_{x\to 0} \left(1+x\right)^\frac{1}{x}=e</math><ref name=":1" />
*<math>\lim_{x\to 0} \left(1+kx\right)^\frac{m}{x}=e^{mk}</math>
*<math>\lim_{x\to+\infty} \left(1+\frac{1}{x}\right)^x=e</math><ref name=":2">((Cite web|url=https://www.pioneermathematics.com/some-important-limits-formula.html|title=SOME IMPORTANT LIMITS - Math Formulas - Mathematics Formulas - Basic Math Formulas|website=www.pioneermathematics.com|access-date=2019-07-31))</ref>
*<math>\lim_{x\to+\infty} \left(1-\frac{1}{x}\right)^x=\frac{1}{e}</math>
*<math>\lim_{x\to+\infty} \left(1+\frac{k}{x}\right)^{mx}=e^{mk} </math><ref name=":3" />
*<math> \lim_{x \to 0} \left(1+ a \left({e^{-x} - 1}\right)\right)^{-\frac{1}{x)) = e^{a} </math>. This limit can be derived from [[#Logarithmic and exponential functions Anchor01|this limit]].

=== Sums, products and composites ===

*<math>\lim_{x \to 0} x e^{-x} = 0 </math>
*<math>\lim_{x \to \infty} x e^{-x} = 0 </math>
*<math>\lim_{x \to 0} \left( \frac{a^x - 1}{x} \right) = \ln{a},</math> for all positive ''a''.<ref name=":6" /><ref name=":2" />
*<math>\lim_{x \to 0} \left( \frac{e^x - 1}{x} \right) = 1 </math>
*<math>\lim_{x \to 0} \left( \frac{e^{ax} - 1}{x} \right) = a </math>

== Logarithmic functions ==

=== Natural logarithms ===
*<math>\lim_{x \to c} \ln{x} = \ln c </math>, due to the continuity of <math>\ln {x} </math>. In particular,
**<math>\lim_{x\to0^+}\log x=-\infty</math>
**<math>\lim_{x\to\infty}\log x=\infty</math>
*<math>\lim_{x\to1}\frac{\ln(x)}{x-1}=1</math>
*<math>\lim_{x\to0}\frac{\ln(x+1)}{x}=1</math><ref name=":2" />
*<math> \lim_{x \to 0} \frac{-\ln\left(1+ a \left({e^{-x} - 1}\right)\right)}{x} = a</math>. This limit follows from [[L'Hôpital's rule]].
*<math>\lim_{x \to 0} x\ln x = 0</math>, hence <math>\lim_{x \to 0} x^x = 1</math>
*<math>\lim_{x \to \infty} \frac{\ln x}{x} = 0</math><ref name=":3" />

=== Logarithms to arbitrary bases ===
For ''b'' > 1,
*<math>\lim_{x \to 0^+} \log_b x = -\infty</math>
*<math>\lim_{x \to \infty} \log_b x = \infty</math>
For ''b'' < 1,
*<math>\lim_{x \to 0^+} \log_b x = \infty</math>
*<math>\lim_{x \to \infty} \log_b x = -\infty</math>
Both cases can be generalized to:
*<math>\lim_{x \to 0^+} \log_b x = -F(b)\infty</math>
*<math>\lim_{x \to \infty} \log_b x = F(b)\infty</math>
where <math>F(x) = 2H(x-1) - 1</math> and <math>H(x)</math> is the [[Heaviside step function]]

==Trigonometric functions==
If <math>x</math> is expressed in radians:

*<math>\lim_{x \to a} \sin x = \sin a</math>
*<math>\lim_{x \to a} \cos x = \cos a</math>

These limits both follow from the continuity of sin and cos.

*<math>\lim_{x \to 0} \frac{\sin x}{x} = 1</math>.<ref name=":2" /><ref name="mit">((cite web| url=https://web.mit.edu/wwmath/calculus/limits/trig.html| title=World Web Math: Useful Trig Limits| publisher=[[Massachusetts Institute of Technology]]
| access-date=2023-03-20))</ref> Or, in general,
**<math>\lim_{x \to 0} \frac{\sin ax}{ax} = 1</math>, for ''a'' not equal to 0.
**<math>\lim_{x \to 0} \frac{\sin ax}{x} = a</math>
**<math>\lim_{x \to 0} \frac{\sin ax}{bx} = \frac{a}{b}</math>, for ''b'' not equal to 0.
*<math>\lim_{x \to \infty} x\sin \left(\frac1x\right) = 1</math>
*<math>\lim_{x \to 0} \frac{1-\cos x}{x} = \lim_{x \to 0} \frac{\cos x - 1}{x} = 0</math><ref name=":6" /><ref name="mit" /><ref>((cite web| url=https://tutorial.math.lamar.edu/classes/calci/prooftrigderiv.aspx| title=Calculus I - Proof of Trig Limits| access-date=2023-03-20))</ref>
*<math>\lim_{x \to 0} \frac{1-\cos x}{x^2} = \frac{1}{2}</math>
*<math>\lim_{x \to n^\pm} \tan \left(\pi x + \frac{\pi}{2}\right) = \mp\infty</math>, for integer ''n''.
*<math>\lim_{x \to 0} \frac{\tan x}{x} = 1</math>. Or, in general,
**<math>\lim_{x \to 0} \frac{\tan ax}{ax} = 1</math>, for ''a'' not equal to 0.
**<math>\lim_{x \to 0} \frac{\tan ax}{bx} = \frac{a}{b}</math>, for ''b'' not equal to 0.
*<math>\lim_{n\to \infty }\ \underbrace{\sin \sin \cdots \sin(x_0)}_n= 0</math>, where ''x''<sub>0</sub> is an arbitrary real number.
*<math>\lim_{n\to \infty }\ \underbrace{\cos \cos \cdots \cos(x_0)}_n= d</math>, where d is the [[Dottie number]]. ''x''<sub>0</sub> 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]]. 

*<math>\lim_{n \to \infty} \sum_{k=1}^n\frac{1}{k}=\infty</math>. This is known as the [[Harmonic series (mathematics)|harmonic series]].<ref name=":3" />
*<math>\lim_{n\to\infty}\left( \sum_{k=1}^{n}\frac{1}{k}-\log n\right)=\gamma</math>. This is the [[Euler Mascheroni constant]].

==Notable special limits==
*<math>\lim_{n\to\infty} \frac{n}{\sqrt[n]{n!))=e</math>
*<math>\lim_{n\to\infty}\left(n!\right)^{1/n}=\infty</math>. This can be proven by considering the inequality <math>e^x \geq \frac{x^n}{n!}</math> at <math>x = n</math>. 
*<math>\lim_{n\to \infty }\, 2^{n} \underbrace{\sqrt{2-\sqrt{2+\sqrt{2+ \dots +\sqrt{2))))}_n= \pi</math>. This can be derived from [[Viète's formula]] for [[pi|((pi))]].

== Limiting behavior ==
===Asymptotic equivalences===

[[Asymptotic analysis|Asymptotic equivalences]], <math>f(x)\sim g(x)</math>, are true if <math>\lim_{x\to\infty}\frac{f(x)}{g(x)}=1</math>. Therefore, they can also be reframed as limits. Some notable asymptotic equivalences include
*<math>\lim_{x\to\infty}\frac{x/\ln x}{\pi(x)}=1</math>, due to the [[prime number theorem]], <math>\pi(x)\sim\frac{x}{\ln x}</math>, where π(x) is the [[Prime-counting function|prime counting function]].
*<math>\lim_{n\to\infty}\frac{\sqrt{2\pi n}\left(\frac{n}{e}\right)^n}{n!}=1</math>, due to [[Stirling's approximation]], <math>n! \sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^n</math>.

===Big O notation===
The behaviour of functions described by [[Big O notation]] can also be described by limits. For example

*<math>f(x)\in\mathcal{O}(g(x))</math> if <math>\limsup_{x\to\infty} \frac{|f(x)|}{g(x)}<\infty</math>

==References==
((Reflist))

((Calculus topics))

[[Category:Limits (mathematics)]]
[[Category:Mathematics-related lists|Limits]]
[[Category:Functions and mappings]]

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