# List of limits

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

$\lim _{x\to c}f(x)=L$ if and only if $\forall \varepsilon >0\ \exists \delta >0:0<|x-c|<\delta \implies |f(x)-L|<\varepsilon$. This is the (ε, δ)-definition of limit.
The limit superior and limit inferior of a sequence are defined as $\limsup _{n\to \infty }x_{n}=\lim _{n\to \infty }\left(\sup _{m\geq n}x_{m}\right)$ and $\liminf _{n\to \infty }x_{n}=\lim _{n\to \infty }\left(\inf _{m\geq n}x_{m}\right)$.
A function, $f(x)$, is said to be continuous at a point, c, if $\lim _{x\to c}f(x)=f(c).$

### Operations on a single known limit

If $\lim _{x\to c}f(x)=L$ then:
* $\lim _{x\to c}\,[f(x)\pm a]=L\pm a$
* $\lim _{x\to c}\,af(x)=aL$
* $\lim _{x\to c}{\frac {1}{f(x)}}={\frac {1}{L}}$ if L is not equal to 0.
* $\lim _{x\to c}\,f(x)^{n}=L^{n}$ if n is a positive integer
* $\lim _{x\to c}\,f(x)^{1 \over n}=L^{1 \over n}$ if n is a positive integer, and if n is even, then L > 0.
In general, if g(x) is continuous at L and $\lim _{x\to c}f(x)=L$ then
* $\lim _{x\to c}g\left(f(x)\right)=g(L)$

### Operations on two known limits

If $\lim _{x\to c}f(x)=L_{1}$ and $\lim _{x\to c}g(x)=L_{2}$ then:
* $\lim _{x\to c}\,[f(x)\pm g(x)]=L_{1}\pm L_{2}$
* $\lim _{x\to c}\,[f(x)g(x)]=L_{1}\cdot L_{2}$
* $\lim _{x\to c}{\frac {f(x)}{g(x)}}={\frac {L_{1}}{L_{2}}}\qquad {\text{ if }}L_{2}\neq 0$

### Limits involving derivatives or infinitesimal changes

In these limits, the infinitesimal change $h$ is often denoted $\Delta x$ or $\delta x$. If $f(x)$ is differentiable at $x$,
* $\lim _{h\to 0}{f(x+h)-f(x) \over h}=f'(x)$. 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,
** $\lim _{h\to 0}{f\circ g(x+h)-f\circ g(x) \over h}=f'[g(x)]g'(x)$. This is the chain rule.
** $\lim _{h\to 0}{f(x+h)g(x+h)-f(x)g(x) \over h}=f'(x)g(x)+f(x)g'(x)$. This is the product rule.
* $\lim _{h\to 0}\left({\frac {f(x+h)}{f(x)}}\right)^{1/h}=\exp \left({\frac {f'(x)}{f(x)}}\right)$
* $\lim _{h\to 0}{\left({f(e^{h}x) \over {f(x)}}\right)^{1/h}}=\exp \left({\frac {xf'(x)}{f(x)}}\right)$
If $f(x)$ and $g(x)$ are differentiable on an open interval containing c, except possibly c itself, and $\lim _{x\to c}f(x)=\lim _{x\to c}g(x)=0{\text{ or }}\pm \infty$, L'Hôpital's rule can be used:
* $\lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}$

### Inequalities

If $f(x)\leq g(x)$ for all x in an interval that contains c, except possibly c itself, and the limit of $f(x)$ and $g(x)$ both exist at c, then $\lim _{x\to c}f(x)\leq \lim _{x\to c}g(x)$
If $\lim _{x\to c}f(x)=\lim _{x\to c}h(x)=L$ and $f(x)\leq g(x)\leq h(x)$ for all x in an open interval that contains c, except possibly c itself, $\lim _{x\to c}g(x)=L.$ 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 x$^a$

* $\lim _{x\to c}a=a$

### Polynomials in x

* $\lim _{x\to c}x=c$
* $\lim _{x\to c}(ax+b)=ac+b$
* $\lim _{x\to c}x^{n}=c^{n}$ if n is a positive integer
* $\lim _{x\to \infty }x/a={\begin{cases}\infty ,&a>0\\{\text{does not exist}},&a=0\\-\infty ,&a<0\end{cases}}$
In general, if $p(x)$ is a polynomial then, by the continuity of polynomials, $\lim _{x\to c}p(x)=p(c)$ This is also true for rational functions, as they are continuous on their domains.

### Functions of the form x$^a$

* $\lim _{x\to c}x^{a}=c^{a}.$ In particular,
** $\lim _{x\to \infty }x^{a}={\begin{cases}\infty ,&a>0\\1,&a=0\\0,&a<0\end{cases}}$
* $\lim _{x\to c}x^{1/a}=c^{1/a}$. In particular,
** $\lim _{x\to \infty }x^{1/a}=\lim _{x\to \infty }{\sqrt[{a}]{x}}=\infty {\text{ for any }}a>0$
* $\lim _{x\to 0^{+}}x^{-n}=\lim _{x\to 0^{+}}{\frac {1}{x^{n}}}=+\infty$
* $\lim _{x\to 0^{-}}x^{-n}=\lim _{x\to 0^{-}}{\frac {1}{x^{n}}}={\begin{cases}-\infty ,&{\text{if }}n{\text{ is odd}}\\+\infty ,&{\text{if }}n{\text{ is even}}\end{cases}}$
* $\lim _{x\to \infty }ax^{-1}=\lim _{x\to \infty }a/x=0{\text{ for any real }}a$

## Exponential functions
### Functions of the form a$^{g(x)}$

* $\lim _{x\to c}e^{x}=e^{c}$, due to the continuity of $e^{x}$
* $\lim _{x\to \infty }a^{x}={\begin{cases}\infty ,&a>1\\1,&a=1\\0,&0<a<1\end{cases}}$
* $\lim _{x\to \infty }a^{-x}={\begin{cases}0,&a>1\\1,&a=1\\\infty ,&0<a<1\end{cases}}$
* $\lim _{x\to \infty }{\sqrt[{x}]{a}}=\lim _{x\to \infty }{a}^{1/x}={\begin{cases}1,&a>0\\0,&a=0\\{\text{does not exist}},&a<0\end{cases}}$

### Functions of the form x$^{g(x)}$

* $\lim _{x\to \infty }{\sqrt[{x}]{x}}=\lim _{x\to \infty }{x}^{1/x}=1$

### Functions of the form f(x)$^{g(x)}$

* $\lim _{x\to +\infty }\left({\frac {x}{x+k}}\right)^{x}=e^{-k}$
* $\lim _{x\to 0}\left(1+x\right)^{\frac {1}{x}}=e$
* $\lim _{x\to 0}\left(1+kx\right)^{\frac {m}{x}}=e^{mk}$
* $\lim _{x\to +\infty }\left(1+{\frac {1}{x}}\right)^{x}=e$
* $\lim _{x\to +\infty }\left(1-{\frac {1}{x}}\right)^{x}={\frac {1}{e}}$
* $\lim _{x\to +\infty }\left(1+{\frac {k}{x}}\right)^{mx}=e^{mk}$
* $\lim _{x\to 0}\left(1+a\left({e^{-x}-1}\right)\right)^{-{\frac {1}{x}}}=e^{a}$. This limit can be derived from this limit.

### Sums, products and composites

* $\lim _{x\to 0}xe^{-x}=0$
* $\lim _{x\to \infty }xe^{-x}=0$
* $\lim _{x\to 0}\left({\frac {a^{x}-1}{x}}\right)=\ln {a},$ for all positive a.
* $\lim _{x\to 0}\left({\frac {e^{x}-1}{x}}\right)=1$
* $\lim _{x\to 0}\left({\frac {e^{ax}-1}{x}}\right)=a$

## Logarithmic functions
### Natural logarithms

* $\lim _{x\to c}\ln {x}=\ln c$, due to the continuity of $\ln {x}$. In particular,
** $\lim _{x\to 0^{+}}\log x=-\infty$
** $\lim _{x\to \infty }\log x=\infty$
* $\lim _{x\to 1}{\frac {\ln(x)}{x-1}}=1$
* $\lim _{x\to 0}{\frac {\ln(x+1)}{x}}=1$
* $\lim _{x\to 0}{\frac {-\ln \left(1+a\left({e^{-x}-1}\right)\right)}{x}}=a$. This limit follows from L'Hôpital's rule.
* $\lim _{x\to 0}x\ln x=0$, hence $\lim _{x\to 0}x^{x}=1$
* $\lim _{x\to \infty }{\frac {\ln x}{x}}=0$

### Logarithms to arbitrary bases

For b > 1,
* $\lim _{x\to 0^{+}}\log _{b}x=-\infty$
* $\lim _{x\to \infty }\log _{b}x=\infty$
For b < 1,
* $\lim _{x\to 0^{+}}\log _{b}x=\infty$
* $\lim _{x\to \infty }\log _{b}x=-\infty$
Both cases can be generalized to:
* $\lim _{x\to 0^{+}}\log _{b}x=-F(b)\infty$
* $\lim _{x\to \infty }\log _{b}x=F(b)\infty$
where $F(x)=2H(x-1)-1$ and $H(x)$ is the Heaviside step function

## Trigonometric functions

If $x$ is expressed in radians:
* $\lim _{x\to a}\sin x=\sin a$
* $\lim _{x\to a}\cos x=\cos a$
These limits both follow from the continuity of sin and cos.
* $\lim _{x\to 0}{\frac {\sin x}{x}}=1$. Or, in general,
** $\lim _{x\to 0}{\frac {\sin ax}{ax}}=1$, for a not equal to 0.
** $\lim _{x\to 0}{\frac {\sin ax}{x}}=a$
** $\lim _{x\to 0}{\frac {\sin ax}{bx}}={\frac {a}{b}}$, for b not equal to 0.
* $\lim _{x\to \infty }x\sin \left({\frac {1}{x}}\right)=1$
* $\lim _{x\to 0}{\frac {1-\cos x}{x}}=\lim _{x\to 0}{\frac {\cos x-1}{x}}=0$
* $\lim _{x\to 0}{\frac {1-\cos x}{x^{2}}}={\frac {1}{2}}$
* $\lim _{x\to n^{\pm }}\tan \left(\pi x+{\frac {\pi }{2}}\right)=\mp \infty$, for integer n.
* $\lim _{x\to 0}{\frac {\tan x}{x}}=1$. Or, in general,
** $\lim _{x\to 0}{\frac {\tan ax}{ax}}=1$, for a not equal to 0.
** $\lim _{x\to 0}{\frac {\tan ax}{bx}}={\frac {a}{b}}$, for b not equal to 0.
* $\lim _{n\to \infty }\ \underbrace {\sin \sin \cdots \sin(x_{0})} _{n}=0$, where x$_0$ is an arbitrary real number.
* $\lim _{n\to \infty }\ \underbrace {\cos \cos \cdots \cos(x_{0})} _{n}=d$, where d is the Dottie number. x$_0$ 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.
* $\lim _{n\to \infty }\sum _{k=1}^{n}{\frac {1}{k}}=\infty$. This is known as the harmonic series.
* $\lim _{n\to \infty }\left(\sum _{k=1}^{n}{\frac {1}{k}}-\log n\right)=\gamma$. This is the Euler Mascheroni constant.

## Notable special limits

* $\lim _{n\to \infty }{\frac {n}{\sqrt[{n}]{n!}}}=e$
* $\lim _{n\to \infty }\left(n!\right)^{1/n}=\infty$. This can be proven by considering the inequality $e^{x}\geq {\frac {x^{n}}{n!}}$ at $x=n$.
* $\lim _{n\to \infty }\,2^{n}\underbrace {\sqrt {2-{\sqrt {2+{\sqrt {2+\dots +{\sqrt {2}}}}}}}} _{n}=\pi$. This can be derived from Viète's formula for π.

## Limiting behavior
### Asymptotic equivalences

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

### Big O notation

The behaviour of functions described by Big O notation can also be described by limits. For example
* $f(x)\in {\mathcal {O}}(g(x))$ if $\limsup _{x\to \infty }{\frac {|f(x)|}{g(x)}}<\infty$