List of limits explained

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} f(x) = L if and only if This is the (ε, δ)-definition of limit.

The limit superior and limit inferior of a sequence are defined as

\limsup _{n} x _{n} = \lim _{n} \left(\sup _{m} x _{m\right)} and \liminf _{n} x _{n} = \lim _{n} \left( inf _{m} x _{m\right)} .

A function,

f(x) , is said to be continuous at a point, c, if $$\backslash lim\_\; f(x)\; =\; f(c).$$

Operations on a single known limit

If

\lim _{x} f(x) = L then:\lim _{x} [f(x) \pm a] = L \pm a \lim _{x} a f(x) = a L ^{[1] [2] [3]} \lim _{x} (1)/(f(x)) = (1 L)/() ^[4] if L is not equal to 0.\lim _{x} f(x) ^{n} = L ^{n} if n is a positive integer\lim _{x} f(x) ^{1} = L ^{1} 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} f(x) = L then\lim _{x} g\left(f(x)\right) = g(L)

Operations on two known limits

If

\lim _{x} f(x) = L _{1} and \lim _{x} g(x) = L _{2} then:\lim _{x} [f(x) \pm g(x)] = L _{1} \pm L _{2} \lim _{x} [f(x)g(x)] = L _{1} ⋅ L _{2} \lim _{x} (f(x))/(g(x)) = (L _{1})/(L _{2}) if L _{2} \ne 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} {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} {f\circ g(x + h) - f\circ g(x)\over h} = f'[g(x)]g'(x) . This is the chain rule.\lim _{h} {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} \left( (f(x + h))/(f(x)) \right) ^{1/h} = \exp\left( (f'(x))/(f(x)) \right) \lim _{h} {\left({f(e ^{h} x)\over{f(x)}}\right) ^{1/h} } = \exp\left( (x f'(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} f(x) = \lim _{x} g(x) = 0 or \pm inf ty , L'Hôpital's rule can be used:\lim _{x} (f(x))/(g(x)) = \lim _{x} (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^[5] $$\backslash lim\_f(x)\backslash leq\; \backslash lim\_g(x)$$

If

\lim _{x} f(x) = \lim _{x} 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,$$\backslash lim\_\; g(x)\; =\; L.$$ 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} a = a

Polynomials in x

\lim _{x} x = c \lim _{x} (ax + b) = ac + b \lim _{x} x ^{n} = c ^{n} if n is a positive integer\lim _{x\toinfty} x/a = \begin{cases} inf ty, & a > 0 \\ doesnotexist , & a = 0 \\ -inf ty, & a < 0 \end{cases}

In general, if

p(x) is a polynomial then, by the continuity of polynomials, $$\backslash lim\_\; p(x)\; =\; p(c)$$ This is also true for rational functions, as they are continuous on their domains.

Functions of the form x^a

\lim _{x\to} x ^{a=c} ^{a.} In particular,\lim _{x\toinfty} x ^{a=\begin{cases}} inf ty, & a > 0 \ 1 , & a = 0 \ 0 , & a < 0 \end{cases} \lim _{x\to} x ^{1/a} = c ^{1/a} . In particular,\lim _{x\toinfty} x ^{1/a} = \lim _{x\toinfty} \sqrt[a]{x} = inf ty forany a > 0 ^[6] \lim _{x \to 0 ^{+}} x ^{-n} = \lim _{x \to 0 ^{+}} (1)/(x ^{n}) = +inf ty \lim _{x \to 0 ^{-}} x ^{-n} = \lim _{x \to 0 ^{-}} (1)/(x ^{n}) = \begin{cases} -inf ty, & if n isodd \\ +inf ty, & if n iseven \end{cases} \lim _{x\toinfty} ax ^{-1} = \lim _{x\toinfty} a/x =0 foranyreal a

Exponential functions

Functions of the form a^g(x)

\lim _{x} e ^{x} = e ^{c} , due to the continuity of e ^{x} \lim _{x\toinfty} a ^{x=\begin{cases}} inf ty, & a > 1 \ 1 , & a = 1 \ 0 , & 0 < a < 1 \end{cases} \lim _{x\toinfty} a ^{-x} = \begin{cases} 0 , & a > 1 \ 1 , & a = 1 \ inf ty, & 0 < a < 1 \end{cases} \lim _{x\toinfty} \sqrt[x]{a} = \lim _{x\toinfty} {a} ^{1/x} = \begin{cases} 1 , & a > 0 \\ 0 , & a = 0 \\ doesnotexist , & a < 0 \end{cases}

Functions of the form x^g(x)

\lim _{x\toinfty} \sqrt[x]{x} = \lim _{x\toinfty} {x} ^{1/x} =1

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

\lim _{x\to+infty} \left( (x)/(x + k) \right) ^{x=e} ^{-k} \lim _{x\to} \left( 1+ x\right) ^{(1)/(x)} = e \lim _{x\to} \left( 1+ kx\right) ^{(m)/(x)} = e ^{mk} \lim _{x\to+infty} \left( 1+ (1)/(x) \right) ^{x=e} ^[7] \lim _{x\to+infty} \left( 1- (1)/(x) \right) ^{x = (1)/(e)} \lim _{x\to+infty} \left( 1+ (k)/(x) \right) ^{mx} = e ^{mk} \lim _{x} \left( 1+ a \left({e ^{-x} - 1 }\right)\right) ^{- (1)/(x)} = e ^{a} . This limit can be derived from this limit.

Sums, products and composites

\lim _{x} x e ^{-x} = 0 \lim _{x} x e ^{-x} = 0 \lim _{x} \left( (a ^{x} -1)/(x) \right) = ln {a}, for all positive a.\lim _{x} \left( (e ^{x} -1)/(x) \right) = 1 \lim _{x} \left( (e ^{ax} -1)/(x) \right) = a

Logarithmic functions

Natural logarithms

\lim _{x} ln {x} = ln c , due to the continuity of ln {x} . In particular,\lim _{x\to 0 ^{+}} log x =-inf ty \lim _{x\toinfty} log x =inf ty \lim _{x\to1} (ln (x))/(x -1) =1 \lim _{x\to0} (ln (x +1 ))/(x) =1 \lim _{x} (-ln \left( 1+ a \left({e ^{-x} -1)/(\right)\right)}{x}) = a . This limit follows from L'Hôpital's rule.\lim _{x} x ln x = 0 , hence \lim _{x} x ^{x} = 1 \lim _{x} (ln x)/(x) = 0

Logarithms to arbitrary bases

For b > 1,

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

Trigonometric functions

If

x is expressed in radians:\lim _{x} \sin x = \sin a \lim _{x} \cos x = \cos a

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

\lim _{x} (\sin x)/(x) = 1 .^[8] Or, in general,\lim _{x} (\sin ax)/(ax) = 1 , for a not equal to 0.\lim _{x} (\sin ax)/(x) = a \lim _{x} (\sin ax)/(bx) = (a)/(b) , for b not equal to 0.\lim _{x} x\sin \left( (1 x\right))/(=) 1 \lim _{x} (1- \cos x)/(x) = \lim _{x} (\cos x -1)/(x) = 0 ^[9] \lim _{x} (1- \cos x)/(x ^{2}) = (1)/(2) \lim _{x \to n ^{\pm}} \tan \left(\pi x + (\pi)/(2) \right) = \mp inf ty , for integer n.\lim _{x} (\tan x)/(x) = 1 . Or, in general,\lim _{x} (\tan ax)/(ax) = 1 , for a not equal to 0.\lim _{x} (\tan ax)/(bx) = (a)/(b) , for b not equal to 0.\lim _{n\to} \underbrace{\sin \sin … \sin(x _{0)}} _{n=} 0 , where x₀ is an arbitrary real number.\lim _{n\to} \underbrace{\cos \cos … \cos(x _{0)}} _{n=} d , 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.

\lim _{n} \sum ^{n (1)/(k)} _{k =1} =inf ty . This is known as the harmonic series.\lim _{n\toinfty} \left( \sum ^{n} _{k =1} (1)/(k) -log n\right) = \gamma . This is the Euler Mascheroni constant.

Notable special limits

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

Limiting behavior

Asymptotic equivalences

Asymptotic equivalences,

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

Notes and References

1. Web site: Basic Limit Laws. math.oregonstate.edu. 2019-07-31.
2. Web site: Limits Cheat Sheet - Symbolab. www.symbolab.com. en. 2019-07-31.
3. Web site: Section 2.3: Calculating Limits using the Limit Laws.
4. Web site: Limits and Derivatives Formulas.
5. Web site: Limits Theorems. archives.math.utk.edu. 2019-07-31.
6. Web site: Some Special Limits. www.sosmath.com. 2019-07-31.
7. Web site: SOME IMPORTANT LIMITS - Math Formulas - Mathematics Formulas - Basic Math Formulas. www.pioneermathematics.com. 2019-07-31.
8. Web site: World Web Math: Useful Trig Limits. Massachusetts Institute of Technology. 2023-03-20.
9. Web site: Calculus I - Proof of Trig Limits. 2023-03-20.