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[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]

Inequalities[edit]

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

Sums[edit]

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

References[edit]

  1. ^ a b c d e f g h i j "Basic Limit Laws". math.oregonstate.edu. Retrieved 2019-07-31.
  2. ^ a b c d e f g h i j k l "Limits Cheat Sheet - Symbolab". www.symbolab.com. 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". archives.math.utk.edu. Retrieved 2019-07-31.
  6. ^ a b c d e "Some Special Limits". www.sosmath.com. Retrieved 2019-07-31.
  7. ^ a b c d "SOME IMPORTANT LIMITS - Math Formulas - Mathematics Formulas - Basic Math Formulas". www.pioneermathematics.com. Retrieved 2019-07-31.