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

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, f(x), is said to be continuous at a point, c, if

Operations on a single known limit

If then: In general, if g(x) is continuous at L and then

Operations on two known limits

If and then:

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

Inequalities

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

Polynomials in x

In general, if p(x) is a polynomial then, by the continuity of polynomials, This is also true for rational functions, as they are continuous on their domains.

Functions of the form xa

Exponential functions

Functions of the form ag(x)

Functions of the form xg(x)

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

Sums, products and composites

Logarithmic functions

Natural logarithms

Logarithms to arbitrary bases

For b > 1, For b < 1, Both cases can be generalized to: where and H(x) is the Heaviside step function

Trigonometric functions

If x is expressed in radians: These limits both follow from the continuity of sin and cos.

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.

Notable special limits

Limiting behavior

Asymptotic equivalences

Asymptotic equivalences,, are true if. 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