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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 =

\lim_{x \to c} f(x) = L if and only if {{nowrap|\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} \, a f(x) = a L{{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}}{{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}}{{Cite web|url=http://faculty.up.edu/wootton/Calc1/Section2.3.pdf|title=Section 2.3: Calculating Limits using the Limit Laws}}
  • \lim_{x \to c} \frac{1}{f(x)}= \frac1L{{Cite web|url=https://www.mathportal.org/formulas/limits_and_derivatives_formulas.pdf|title=Limits and Derivatives Formulas}} 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 \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 \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{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 \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{{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}}

\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<sup>a</sup>''

  • \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<sup>a</sup>'' =

  • \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 {{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}}
  • \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''<sup>''g''(''x'')</sup> =

  • \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''<sup>''g''(''x'')</sup> =

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

= Functions of the form ''f''(''x'')<sup>''g''(''x'')</sup> =

  • \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{{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}}
  • \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} x e^{-x} = 0
  • \lim_{x \to \infty} x e^{-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\to0^+}\log x=-\infty
  • \lim_{x\to\infty}\log x=\infty
  • \lim_{x\to1}\frac{\ln(x)}{x-1}=1
  • \lim_{x\to0}\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.{{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}} 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(\frac1x\right) = 1
  • \lim_{x \to 0} \frac{1-\cos x}{x} = \lim_{x \to 0} \frac{\cos x - 1}{x} = 0{{cite web| url=https://tutorial.math.lamar.edu/classes/calci/prooftrigderiv.aspx| title=Calculus I - Proof of Trig Limits| access-date=2023-03-20}}
  • \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 x0 is an arbitrary real number.
  • \lim_{n\to \infty }\ \underbrace{\cos \cos \cdots \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 \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 {{pi}}.

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

=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

References

{{Reflist}}

{{Calculus topics}}

Category:Limits (mathematics)

Limits

Category:Functions and mappings