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.
lim
∀ ε > 0
< ε
This is the (ε, δ)-definition of limit.
The limit superior and limit inferior of a sequence are defined as
lim sup
lim inf
inf
A function,
, is said to be continuous at a point, c, if
then: In general, if g(x) is continuous at L and
then: In these limits, the infinitesimal change
are differentiable on an open interval containing c, except possibly c itself, and
, L'Hôpital's rule can be used: If
for all x in an interval that contains c, except possibly c itself, and the limit of
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. 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] For b > 1, For b < 1, Both cases can be generalized to: where
is the Heaviside step function If
is expressed in radians: These limits both follow from the continuity of sin and cos.
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.
Asymptotic equivalences,
Therefore, they can also be reframed as limits.
Some notable asymptotic equivalences include The behaviour of functions described by Big O notation can also be described by limits.