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 $\forall ε > 0 \exists δ > 0 : 0 < | x - c | < δ ⟹ | f (x) - L | < ε$ . This is the (ε, δ)-definition of limit.

The limit superior and limit inferior of a sequence are defined as $\underset{n \to \infty}{lim sup} x_{n} = lim_{n \to \infty} (sup_{m \geq n} x_{m})$ and $\underset{n \to \infty}{lim inf} x_{n} = lim_{n \to \infty} (inf_{m \geq n} x_{m})$ .

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:

$lim_{x \to c} [f (x) \pm a] = L \pm a$
$lim_{x \to c} a f (x) = a L$
$lim_{x \to c} \frac{1}{f (x)} = \frac{1}{L}$ if L is not equal to 0.
$lim_{x \to c} f (x)^{n} = L^{n}$ if n is a positive integer
$lim_{x \to c} f (x)^{\frac{1}{n}} = L^{\frac{1}{n}}$ 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

$lim_{x \to c} g (f (x)) = g (L)$

Operations on two known limits

If $lim_{x \to c} f (x) = L_{1}$ and $lim_{x \to c} g (x) = L_{2}$ then:

$lim_{x \to c} [f (x) \pm g (x)] = L_{1} \pm L_{2}$
$lim_{x \to c} [f (x) g (x)] = L_{1} \cdot L_{2}$
$lim_{x \to c} \frac{f (x)}{g (x)} = \frac{L_{1}}{L_{2}} if L_{2} \neq 0$

Limits involving derivatives or infinitesimal changes

In these limits, the infinitesimal change $h$ is often denoted $Δ x$ or $δ x$ . If $f (x)$ is differentiable at $x$ ,

limh→0f(x+h)−f(x)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,
- $lim_{h \to 0} \frac{f \circ g (x + h) - f \circ g (x)}{h} = f^{'} [g (x)] g^{'} (x)$ . This is the chain rule.
- $lim_{h \to 0} \frac{f (x + h) g (x + h) - f (x) g (x)}{h} = f^{'} (x) g (x) + f (x) g^{'} (x)$ . This is the product rule.
$lim_{h \to 0} {(\frac{f (x + h)}{f (x)})}^{1 / h} = \exp (\frac{f^{'} (x)}{f (x)})$
$lim_{h \to 0} {(\frac{f (e^{h} x)}{f (x)})}^{1 / h} = \exp (\frac{x f^{'} (x)}{f (x)})$

If $f (x)$ and $g (x)$ are differentiable on an open interval containing c, except possibly c itself, and $lim_{x \to c} f (x) = lim_{x \to c} g (x) = 0 or \pm \infty$ , 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

$lim_{x \to c} x = c$
$lim_{x \to c} (a x + b) = a c + b$
$lim_{x \to c} x^{n} = c^{n}$ if n is a positive integer
$lim_{x \to \infty} x / a = {\begin{matrix} \infty, & a > 0 \\ does not exist, & a = 0 \\ - \infty, & a < 0 \end{matrix}$

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

limx→cxa=ca. In particular,
- $lim_{x \to \infty} x^{a} = {\begin{matrix} \infty, & a > 0 \\ 1, & a = 0 \\ 0, & a < 0 \end{matrix}$
limx→cx1/a=c1/a. In particular,
- $lim_{x \to \infty} x^{1 / a} = lim_{x \to \infty} \sqrt[a]{x} = \infty for any a > 0$
$lim_{x \to 0^{+}} x^{- n} = lim_{x \to 0^{+}} \frac{1}{x^{n}} = + \infty$
$lim_{x \to 0^{-}} x^{- n} = lim_{x \to 0^{-}} \frac{1}{x^{n}} = {\begin{matrix} - \infty, & if n is odd \\ + \infty, & if n is even \end{matrix}$
$lim_{x \to \infty} a x^{- 1} = lim_{x \to \infty} a / x = 0 for any real a$

Exponential functions

Functions of the form a^g(x)

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

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)

$lim_{x \to + \infty} {(\frac{x}{x + k})}^{x} = e^{- k}$
$lim_{x \to 0} {(1 + x)}^{\frac{1}{x}} = e$
$lim_{x \to 0} {(1 + k x)}^{\frac{m}{x}} = e^{m k}$
$lim_{x \to + \infty} {(1 + \frac{1}{x})}^{x} = e$
$lim_{x \to + \infty} {(1 - \frac{1}{x})}^{x} = \frac{1}{e}$
$lim_{x \to + \infty} {(1 + \frac{k}{x})}^{m x} = e^{m k}$
$lim_{x \to 0} {(1 + a (e^{- x} - 1))}^{- \frac{1}{x}} = e^{a}$ . This limit can be derived from this limit.

Sums, products and composites

$lim_{x \to 0} x e^{- x} = 0$
$lim_{x \to \infty} x e^{- x} = 0$
$lim_{x \to 0} (\frac{a^{x} - 1}{x}) = \ln a,$ for all positive a.
$lim_{x \to 0} (\frac{e^{x} - 1}{x}) = 1$
$lim_{x \to 0} (\frac{e^{a x} - 1}{x}) = a$

Logarithmic functions

Natural logarithms

limx→cln⁡x=ln⁡c, due to the continuity of ln⁡x. In particular,
- $lim_{x \to 0^{+}} \log x = - \infty$
- $lim_{x \to \infty} \log x = \infty$
$lim_{x \to 1} \frac{\ln (x)}{x - 1} = 1$
$lim_{x \to 0} \frac{\ln (x + 1)}{x} = 1$
$lim_{x \to 0} \frac{- \ln (1 + a (e^{- x} - 1))}{x} = a$ . 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 $F (x) = 2 H (x - 1) - 1$ 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.

limx→0sin⁡xx=1. Or, in general,
- $lim_{x \to 0} \frac{\sin a x}{a x} = 1$ , for a not equal to 0.
- $lim_{x \to 0} \frac{\sin a x}{x} = a$
- $lim_{x \to 0} \frac{\sin a x}{b x} = \frac{a}{b}$ , for b not equal to 0.
$lim_{x \to \infty} x \sin (\frac{1}{x}) = 1$
$lim_{x \to 0} \frac{1 - \cos x}{x} = lim_{x \to 0} \frac{\cos x - 1}{x} = 0$
$lim_{x \to 0} \frac{1 - \cos x}{x^{2}} = \frac{1}{2}$
$lim_{x \to n^{\pm}} \tan (π x + \frac{π}{2}) = \mp \infty$ , for integer n.
limx→0tan⁡xx=1. Or, in general,
- $lim_{x \to 0} \frac{\tan a x}{a x} = 1$ , for a not equal to 0.
- $lim_{x \to 0} \frac{\tan a x}{b x} = \frac{a}{b}$ , for b not equal to 0.
$lim_{n \to \infty} \underset{n}{\underset{⏟}{\sin \sin \dots \sin (x_{0})}} = 0$ , where x₀ is an arbitrary real number.
$lim_{n \to \infty} \underset{n}{\underset{⏟}{\cos \cos \dots \cos (x_{0})}} = d$ , 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.

$lim_{n \to \infty} \sum_{k = 1}^{n} \frac{1}{k} = \infty$ . This is known as the harmonic series.
$lim_{n \to \infty} (\sum_{k = 1}^{n} \frac{1}{k} - \log n) = γ$ . This is the Euler Mascheroni constant.

Notable special limits

$lim_{n \to \infty} \frac{n}{\sqrt[n]{n!}} = e$
$lim_{n \to \infty} {(n!)}^{1 / n} = \infty$ . This can be proven by considering the inequality $e^{x} \geq \frac{x^{n}}{n!}$ at $x = n$ .
$lim_{n \to \infty} 2^{n} \underset{n}{\underset{⏟}{\sqrt{2 - \sqrt{2 + \sqrt{2 + \dots + \sqrt{2}}}}}} = π$ . 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

$lim_{x \to \infty} \frac{x / \ln x}{π (x)} = 1$ , due to the prime number theorem, $π (x) \sim \frac{x}{\ln x}$ , where π(x) is the prime counting function.
$lim_{n \to \infty} \frac{\sqrt{2 π n} {(\frac{n}{e})}^{n}}{n!} = 1$ , due to Stirling's approximation, $n! \sim \sqrt{2 π n} {(\frac{n}{e})}^{n}$ .

Big O notation

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

$f (x) \in 𝒪 (g (x))$ if $\underset{x \to \infty}{lim sup} \frac{| f (x) |}{g (x)} < \infty$

References

1 2 3 4 5 6 7 8 9 10 "Basic Limit Laws". math.oregonstate.edu. Archived from the original on 2019-07-30. Retrieved 2019-07-31.
1 2 3 4 5 6 7 8 9 10 11 12 "Limits Cheat Sheet - Symbolab". www.symbolab.com. Retrieved 2019-07-31.
1 2 3 4 5 6 7 8 "Section 2.3: Calculating Limits using the Limit Laws" (PDF).
1 2 3 "Limits and Derivatives Formulas" (PDF).
1 2 3 4 5 6 "Limits Theorems". archives.math.utk.edu. Retrieved 2019-07-31.
1 2 3 4 5 "Some Special Limits". www.sosmath.com. Retrieved 2019-07-31.
1 2 3 4 "SOME IMPORTANT LIMITS - Math Formulas - Mathematics Formulas - Basic Math Formulas". www.pioneermathematics.com. Archived from the original on 2019-07-31. Retrieved 2019-07-31.
1 2 "World Web Math: Useful Trig Limits". Massachusetts Institute of Technology. Retrieved 2023-03-20.
↑ "Calculus I - Proof of Trig Limits". Retrieved 2023-03-20.

[:0-1] 1 2 3 4 5 6 7 8 9 10 "Basic Limit Laws". math.oregonstate.edu. Archived from the original on 2019-07-30. Retrieved 2019-07-31.

[:1-2] 1 2 3 4 5 6 7 8 9 10 11 12 "Limits Cheat Sheet - Symbolab". www.symbolab.com. Retrieved 2019-07-31.

[:5-3] 1 2 3 4 5 6 7 8 "Section 2.3: Calculating Limits using the Limit Laws" (PDF).

[:6-4] 1 2 3 "Limits and Derivatives Formulas" (PDF).

[:4-5] 1 2 3 4 5 6 "Limits Theorems". archives.math.utk.edu. Retrieved 2019-07-31.

[:3-6] 1 2 3 4 5 "Some Special Limits". www.sosmath.com. Retrieved 2019-07-31.

[:2-7] 1 2 3 4 "SOME IMPORTANT LIMITS - Math Formulas - Mathematics Formulas - Basic Math Formulas". www.pioneermathematics.com. Archived from the original on 2019-07-31. Retrieved 2019-07-31.

[mit-8] 1 2 "World Web Math: Useful Trig Limits". Massachusetts Institute of Technology. Retrieved 2023-03-20.

[9] "Calculus I - Proof of Trig Limits". Retrieved 2023-03-20.