List of limits

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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

[math]\displaystyle{ \lim_{x \to c} f(x) = L }[/math] if and only if [math]\displaystyle{ \forall \varepsilon\gt 0\ \exists \delta\gt 0 : 0 \lt |x-c| \lt \delta \implies |f(x)-L| \lt \varepsilon }[/math]. This is the (ε, δ)-definition of limit.

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

A function, [math]\displaystyle{ f(x) }[/math], is said to be continuous at a point, c, if [math]\displaystyle{ \lim_{x \to c} f(x) = f(c). }[/math]

Operations on a single known limit

If [math]\displaystyle{ \lim_{x \to c} f(x) = L }[/math] then:

[math]\displaystyle{ \lim_{x \to c} \, [f(x) \pm a] = L \pm a }[/math]
[math]\displaystyle{ \lim_{x \to c} \, a f(x) = a L }[/math]^[1]^[2]^[3]
[math]\displaystyle{ \lim_{x \to c} \frac{1}{f(x)}= \frac1L }[/math]^[4] if L is not equal to 0.
[math]\displaystyle{ \lim_{x \to c} \, f(x)^n = L^n }[/math] if n is a positive integer^[1]^[2]^[3]
[math]\displaystyle{ \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.^[1]^[3]

In general, if g(x) is continuous at L and [math]\displaystyle{ \lim_{x \to c} f(x) = L }[/math] then

[math]\displaystyle{ \lim_{x \to c} g\left(f(x)\right) =g(L) }[/math]^[1]^[2]

Operations on two known limits

If [math]\displaystyle{ \lim_{x \to c} f(x) = L_1 }[/math] and [math]\displaystyle{ \lim_{x \to c} g(x) = L_2 }[/math] then:

[math]\displaystyle{ \lim_{x \to c} \, [f(x) \pm g(x)] = L_1 \pm L_2 }[/math]^[1]^[2]^[3]
[math]\displaystyle{ \lim_{x \to c} \, [f(x)g(x)] = L_1 \cdot L_2 }[/math]^[1]^[2]^[3]
[math]\displaystyle{ \lim_{x \to c} \frac{f(x)}{g(x)} = \frac{L_1}{L_2} \qquad \text{ if } L_2 \ne 0 }[/math]^[1]^[2]^[3]

Limits involving derivatives or infinitesimal changes

In these limits, the infinitesimal change [math]\displaystyle{ h }[/math] is often denoted [math]\displaystyle{ \Delta x }[/math] or [math]\displaystyle{ \delta x }[/math]. If [math]\displaystyle{ f(x) }[/math]is differentiable at [math]\displaystyle{ x }[/math],

[math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \lim_{h \to 0} \left(\frac{f(x+h)}{f(x)}\right)^{1/h} = \exp\left(\frac{f'(x)}{f(x)}\right) }[/math]
[math]\displaystyle{ \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]\displaystyle{ f(x) }[/math] and [math]\displaystyle{ g(x) }[/math] are differentiable on an open interval containing c, except possibly c itself, and [math]\displaystyle{ \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]\displaystyle{ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} }[/math]^[2]

Inequalities

If [math]\displaystyle{ f(x)\leq g(x) }[/math] for all x in an interval that contains c, except possibly c itself, and the limit of [math]\displaystyle{ f(x) }[/math] and [math]\displaystyle{ g(x) }[/math] both exist at c, then^[5] [math]\displaystyle{ \lim_{x\to c}f(x)\leq \lim_{x\to c}g(x) }[/math]

If [math]\displaystyle{ \lim_{x \to c} f(x) = \lim_{x \to c} h(x) = L }[/math] and [math]\displaystyle{ f(x)\leq g(x)\leq h(x) }[/math]for all x in an open interval that contains c, except possibly c itself, [math]\displaystyle{ \lim_{x \to c} g(x) = L. }[/math] 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 x^a

[math]\displaystyle{ \lim_{x \to c} a = a }[/math]^[1]^[2]^[3]

Polynomials in x

[math]\displaystyle{ \lim_{x \to c} x = c }[/math]^[1]^[2]^[3]
[math]\displaystyle{ \lim_{x \to c} (ax + b) = ac + b }[/math]
[math]\displaystyle{ \lim_{x \to c} x^n = c^n }[/math] if n is a positive integer^[5]
[math]\displaystyle{ \lim_{x\to\infty} x/a = \begin{cases} \infty, & a \gt 0 \\ \text{does not exist}, & a = 0 \\ -\infty, & a \lt 0 \end{cases} }[/math]

In general, if [math]\displaystyle{ p(x) }[/math]is a polynomial then, by the continuity of polynomials,^[5] [math]\displaystyle{ \lim_{x \to c} p(x) = p(c) }[/math] This is also true for rational functions, as they are continuous on their domains.^[5]

Functions of the form x^a

[math]\displaystyle{ \lim_{x\to c}x^a=c^a. }[/math]^[5] In particular,
- [math]\displaystyle{ \lim_{x\to\infty}x^a=\begin{cases} \infty, & a \gt 0 \\ 1, & a = 0 \\ 0, & a \lt 0 \end{cases} }[/math]
[math]\displaystyle{ \lim_{x\to c}x^{1/a}=c^{1/a} }[/math].^[5] In particular,
- [math]\displaystyle{ \lim_{x\to\infty} x^{1/a}=\lim_{x\to\infty}\sqrt[a]{x}= \infty \text{ for any } a \gt 0 }[/math]^[6]
[math]\displaystyle{ \lim_{x \to 0^+} x^{-n} =\lim_{x \to 0^+} \frac{1}{x^n}= +\infty }[/math]
[math]\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} }[/math]
[math]\displaystyle{ \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^g(x)

[math]\displaystyle{ \lim_{x \to c} e^{x} = e^c }[/math], due to the continuity of [math]\displaystyle{ e^{x} }[/math]
[math]\displaystyle{ \lim_{x\to\infty}a^x=\begin{cases} \infty, & a \gt 1 \\ 1, & a = 1 \\ 0, & 0 \lt a \lt 1 \end{cases} }[/math]
[math]\displaystyle{ \lim_{x\to\infty}a^{-x}=\begin{cases} 0, & a \gt 1 \\ 1, & a = 1 \\ \infty, & 0 \lt a \lt 1 \end{cases} }[/math]^[6]
[math]\displaystyle{ \lim_{x\to\infty}\sqrt[x]{a}=\lim_{x\to\infty}{a}^{1/x}=\begin{cases} 1, & a \gt 0 \\ 0, & a = 0 \\ \text{does not exist}, & a \lt 0 \end{cases} }[/math]

Functions of the form x^g(x)

[math]\displaystyle{ \lim_{x\to\infty}\sqrt[x]{x}=\lim_{x\to\infty}{x}^{1/x}=1 }[/math]

Functions of the form f(x)^g(x)

[math]\displaystyle{ \lim_{x\to+\infty} \left( \frac{x}{x+k}\right)^x=e^{-k} }[/math]^[2]
[math]\displaystyle{ \lim_{x\to 0} \left(1+x\right)^\frac{1}{x}=e }[/math]^[2]
[math]\displaystyle{ \lim_{x\to 0} \left(1+kx\right)^\frac{m}{x}=e^{mk} }[/math]
[math]\displaystyle{ \lim_{x\to+\infty} \left(1+\frac{1}{x}\right)^x=e }[/math]^[7]
[math]\displaystyle{ \lim_{x\to+\infty} \left(1-\frac{1}{x}\right)^x=\frac{1}{e} }[/math]
[math]\displaystyle{ \lim_{x\to+\infty} \left(1+\frac{k}{x}\right)^{mx}=e^{mk} }[/math]^[6]
[math]\displaystyle{ \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 this limit.

Sums, products and composites

[math]\displaystyle{ \lim_{x \to 0} x e^{-x} = 0 }[/math]
[math]\displaystyle{ \lim_{x \to \infty} x e^{-x} = 0 }[/math]
[math]\displaystyle{ \lim_{x \to 0} \left( \frac{a^x - 1}{x} \right) = \ln{a}, }[/math] for all positive a.^[4]^[7]
[math]\displaystyle{ \lim_{x \to 0} \left( \frac{e^x - 1}{x} \right) = 1 }[/math]
[math]\displaystyle{ \lim_{x \to 0} \left( \frac{e^{ax} - 1}{x} \right) = a }[/math]

Logarithmic functions

Natural logarithms

[math]\displaystyle{ \lim_{x \to c} \ln{x} = \ln c }[/math], due to the continuity of [math]\displaystyle{ \ln {x} }[/math]. In particular,
- [math]\displaystyle{ \lim_{x\to0^+}\log x=-\infty }[/math]
- [math]\displaystyle{ \lim_{x\to\infty}\log x=\infty }[/math]
[math]\displaystyle{ \lim_{x\to1}\frac{\ln(x)}{x-1}=1 }[/math]
[math]\displaystyle{ \lim_{x\to0}\frac{\ln(x+1)}{x}=1 }[/math]^[7]
[math]\displaystyle{ \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]\displaystyle{ \lim_{x \to 0} x\ln x = 0 }[/math], hence [math]\displaystyle{ \lim_{x \to 0} x^x = 1 }[/math]
[math]\displaystyle{ \lim_{x \to \infty} \frac{\ln x}{x} = 0 }[/math]^[6]

Logarithms to arbitrary bases

For b > 1,

[math]\displaystyle{ \lim_{x \to 0^+} \log_b x = -\infty }[/math]
[math]\displaystyle{ \lim_{x \to \infty} \log_b x = \infty }[/math]

For b < 1,

[math]\displaystyle{ \lim_{x \to 0^+} \log_b x = \infty }[/math]
[math]\displaystyle{ \lim_{x \to \infty} \log_b x = -\infty }[/math]

Both cases can be generalized to:

[math]\displaystyle{ \lim_{x \to 0^+} \log_b x = -F(b)\infty }[/math]
[math]\displaystyle{ \lim_{x \to \infty} \log_b x = F(b)\infty }[/math]

where [math]\displaystyle{ F(x) = 2H(x-1) - 1 }[/math] and [math]\displaystyle{ H(x) }[/math] is the Heaviside step function

Trigonometric functions

If [math]\displaystyle{ x }[/math] is expressed in radians:

[math]\displaystyle{ \lim_{x \to a} \sin x = \sin a }[/math]
[math]\displaystyle{ \lim_{x \to a} \cos x = \cos a }[/math]

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

[math]\displaystyle{ \lim_{x \to 0} \frac{\sin x}{x} = 1 }[/math].^[7]^[8] Or, in general,
- [math]\displaystyle{ \lim_{x \to 0} \frac{\sin ax}{ax} = 1 }[/math], for a not equal to 0.
- [math]\displaystyle{ \lim_{x \to 0} \frac{\sin ax}{x} = a }[/math]
- [math]\displaystyle{ \lim_{x \to 0} \frac{\sin ax}{bx} = \frac{a}{b} }[/math], for b not equal to 0.
[math]\displaystyle{ \lim_{x \to \infty} x\sin \left(\frac1x\right) = 1 }[/math]
[math]\displaystyle{ \lim_{x \to 0} \frac{1-\cos x}{x} = \lim_{x \to 0} \frac{\cos x - 1}{x} = 0 }[/math]^[4]^[8]^[9]
[math]\displaystyle{ \lim_{x \to 0} \frac{1-\cos x}{x^2} = \frac{1}{2} }[/math]
[math]\displaystyle{ \lim_{x \to n^\pm} \tan \left(\pi x + \frac{\pi}{2}\right) = \mp\infty }[/math], for integer n.
[math]\displaystyle{ \lim_{x \to 0} \frac{\tan x}{x} = 1 }[/math]. Or, in general,
- [math]\displaystyle{ \lim_{x \to 0} \frac{\tan ax}{ax} = 1 }[/math], for a not equal to 0.
- [math]\displaystyle{ \lim_{x \to 0} \frac{\tan ax}{bx} = \frac{a}{b} }[/math], for b not equal to 0.
[math]\displaystyle{ \lim_{n\to \infty }\ \underbrace{\sin \sin \cdots \sin(x_0)}_n= 0 }[/math], where x₀ is an arbitrary real number.
[math]\displaystyle{ \lim_{n\to \infty }\ \underbrace{\cos \cos \cdots \cos(x_0)}_n= d }[/math], 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.

[math]\displaystyle{ \lim_{n \to \infty} \sum_{k=1}^n\frac{1}{k}=\infty }[/math]. This is known as the harmonic series.^[6]
[math]\displaystyle{ \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]\displaystyle{ \lim_{n\to\infty} \frac{n}{\sqrt[n]{n!}}=e }[/math]
[math]\displaystyle{ \lim_{n\to\infty}\left(n!\right)^{1/n}=\infty }[/math]. This can be proven by considering the inequality [math]\displaystyle{ e^x \geq \frac{x^n}{n!} }[/math] at [math]\displaystyle{ x = n }[/math].
[math]\displaystyle{ \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 $π$ .

Limiting behavior

Asymptotic equivalences

Asymptotic equivalences, [math]\displaystyle{ f(x)\sim g(x) }[/math], are true if [math]\displaystyle{ \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]\displaystyle{ \lim_{x\to\infty}\frac{x/\ln x}{\pi(x)}=1 }[/math], due to the prime number theorem, [math]\displaystyle{ \pi(x)\sim\frac{x}{\ln x} }[/math], where π(x) is the prime counting function.
[math]\displaystyle{ \lim_{n\to\infty}\frac{\sqrt{2\pi n}\left(\frac{n}{e}\right)^n}{n!}=1 }[/math], due to Stirling's approximation, [math]\displaystyle{ 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]\displaystyle{ f(x)\in\mathcal{O}(g(x)) }[/math] if [math]\displaystyle{ \limsup_{x\to\infty} \frac{|f(x)|}{g(x)}\lt \infty }[/math]

References

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Original source: https://en.wikipedia.org/wiki/List of limits. Read more

[:0-1] 1.0 ^1.1 ^1.2 ^1.3 ^1.4 ^1.5 ^1.6 ^1.7 ^1.8 ^1.9 "Basic Limit Laws". https://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/SandS/lHopital/limit_laws.html.

[:1-2] 2.00 ^2.01 ^2.02 ^2.03 ^2.04 ^2.05 ^2.06 ^2.07 ^2.08 ^2.09 ^2.10 ^2.11 "Limits Cheat Sheet - Symbolab" (in en). https://www.symbolab.com/cheat-sheets/Limits#.

[:5-3] 3.0 ^3.1 ^3.2 ^3.3 ^3.4 ^3.5 ^3.6 ^3.7 "Section 2.3: Calculating Limits using the Limit Laws". http://faculty.up.edu/wootton/Calc1/Section2.3.pdf.

[:6-4] 4.0 ^4.1 ^4.2 "Limits and Derivatives Formulas". https://www.mathportal.org/formulas/limits_and_derivatives_formulas.pdf.

[:4-5] 5.0 ^5.1 ^5.2 ^5.3 ^5.4 ^5.5 "Limits Theorems". http://archives.math.utk.edu/visual.calculus/1/limits.18/index.html.

[:3-6] 6.0 ^6.1 ^6.2 ^6.3 ^6.4 "Some Special Limits". http://www.sosmath.com/calculus/sequence/specialim/specialim.html.

[:2-7] 7.0 ^7.1 ^7.2 ^7.3 "SOME IMPORTANT LIMITS - Math Formulas - Mathematics Formulas - Basic Math Formulas". https://www.pioneermathematics.com/some-important-limits-formula.html.

[mit-8] 8.0 ^8.1 "World Web Math: Useful Trig Limits". Massachusetts Institute of Technology. https://web.mit.edu/wwmath/calculus/limits/trig.html.

[9] "Calculus I - Proof of Trig Limits". https://tutorial.math.lamar.edu/classes/calci/prooftrigderiv.aspx.

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Differential calculus	Chain rule Derivative Differential Differential equation Differential operator Implicit differentiation Inverse functions and differentiation L'Hôpital's rule Leibniz's rule Logarithmic derivative Mean value theorem Newton's method Notation Leibniz's notation Newton's notation Product rule Quotient rule Regiomontanus' angle maximization problem Related rates Simplest rules Constant factor rule in differentiation Linearity of differentiation Power rule Sum rule in differentiation Stationary point First derivative test Second derivative test Extreme value theorem Maxima and minima Taylor's theorem
Integral calculus	Antiderivative Arc length Constant of integration Differentiation under the integral sign Fundamental theorem of calculus Integral of secant cubed Integral of the secant function Integration by parts Integration by substitution Tangent half-angle substitution Partial fractions in integration Quadratic integral Proof that 22/7 exceeds π Simplest rules Constant factor rule in integration Linearity of integration Sum rule in integration Trapezoidal rule Trigonometric substitution
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Contents

Limits for general functions

Definitions of limits and related concepts

Operations on a single known limit

Operations on two known limits

Limits involving derivatives or infinitesimal changes

Inequalities

Polynomials and functions of the form x^a

Polynomials in x

Functions of the form x^a

Exponential functions

Functions of the form a^g(x)

Functions of the form x^g(x)

Functions of the form f(x)^g(x)

Sums, products and composites

Logarithmic functions

Natural logarithms

Logarithms to arbitrary bases

Trigonometric functions

Sums

Notable special limits

Limiting behavior

Asymptotic equivalences

Big O notation

References

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List of limits

Limits for general functions

Definitions of limits and related concepts

Operations on a single known limit

Operations on two known limits

Limits involving derivatives or infinitesimal changes

Inequalities

Polynomials and functions of the form xa

Polynomials in x

Functions of the form xa

Exponential functions

Functions of the form ag(x)

Functions of the form xg(x)

Functions of the form f(x)g(x)

Sums, products and composites

Logarithmic functions

Natural logarithms

Logarithms to arbitrary bases

Trigonometric functions

Sums

Notable special limits

Limiting behavior

Asymptotic equivalences

Big O notation

References

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Polynomials and functions of the form x^a

Functions of the form x^a

Functions of the form a^g(x)

Functions of the form x^g(x)

Functions of the form f(x)^g(x)