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 SM

Limits for general functions[edit]

Definitions of limits and related concepts[edit]

if and only if . This is the (ε, δ)-definition of limit.

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

A function, , is said to be continuous at a point, c, if

Operations on a single known limit[edit]

If then:

  • [1][2][3]
  • [4] if L is not equal to 0.
  • if n is a positive integer[1][2][3]
  • 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 then

  • [1][2]

Operations on two known limits[edit]

If and then:

  • [1][2][3]
  • [1][2][3]
  • [1][2][3]

Limits involving derivatives or infinitesimal changes[edit]

In these limits, the infinitesimal change is often denoted or . If is differentiable at ,

  • . 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,
    • . This is the chain rule.
    • . This is the product rule.

If and are differentiable on an open interval containing c, except possibly c itself, and , L'Hôpital's rule can be used:

  • [2]


If for all x in an interval that contains c, except possibly c itself, and the limit of and both exist at c, then[5]

If and for all x in an open interval that contains c, except possibly c itself,

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 xa[edit]

  • [1][2][3]

Polynomials in x[edit]

  • [1][2][3]
  • if n is a positive integer[5]

In general, if is a polynomial then, by the continuity of polynomials,[5]

This is also true for rational functions, as they are continuous on their domains.[5]

Functions of the form xa[edit]

  • [5] In particular,
  • .[5] In particular,
    • [6]

Exponential functions[edit]

Functions of the form ag(x)[edit]

  • , due to the continuity of
  • [6]

Functions of the form xg(x)[edit]

Functions of the form f(x)g(x)[edit]

  • [2]
  • [2]
  • [7]
  • [6]
  • . This limit can be derived from this limit.

Sums, products and composites[edit]

  • for all positive a.[4][7]

Logarithmic functions[edit]

Natural logarithms[edit]

  • , due to the continuity of . In particular,
  • [7]
  • . This limit follows from L'Hôpital's rule.
  • , hence
  • [6]

Logarithms to arbitrary bases[edit]

For b > 1,

For b < 1,

Both cases can be generalized to:

where and is the Heaviside step function

Trigonometric functions[edit]

If is expressed in radians:

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

  • .[7][8] Or, in general,
    • , for a not equal to 0.
    • , for b not equal to 0.
  • [4][8][9]
  • , for integer n.
  • . Or, in general,
    • , for a not equal to 0.
    • , for b not equal to 0.
  • , where x0 is an arbitrary real number.
  • , where d is the Dottie number. x0 can be any arbitrary real number.


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.

  • . This is known as the harmonic series.[6]
  • . This is the Euler Mascheroni constant.

Notable special limits[edit]

  • . This can be proven by considering the inequality at .
  • . This can be derived from Viète's formula for π.

Limiting behavior[edit]

Asymptotic equivalences[edit]

Asymptotic equivalences, , are true if . Therefore, they can also be reframed as limits. Some notable asymptotic equivalences include

  • , due to the prime number theorem, , where π(x) is the prime counting function.
  • , due to Stirling's approximation, .

Big O notation[edit]

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

  • if


  1. ^ a b c d e f g h i j "Basic Limit Laws". Retrieved 2019-07-31.
  2. ^ a b c d e f g h i j k l "Limits Cheat Sheet - Symbolab". Retrieved 2019-07-31.
  3. ^ a b c d e f g h "Section 2.3: Calculating Limits using the Limit Laws" (PDF).
  4. ^ a b c "Limits and Derivatives Formulas" (PDF).
  5. ^ a b c d e f "Limits Theorems". Retrieved 2019-07-31.
  6. ^ a b c d e "Some Special Limits". Retrieved 2019-07-31.
  7. ^ a b c d "SOME IMPORTANT LIMITS - Math Formulas - Mathematics Formulas - Basic Math Formulas". Retrieved 2019-07-31.
  8. ^ a b "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.