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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.
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
If then:
In general, if g(x) is continuous at L and then
If and then:
In these limits, the infinitesimal change is often denoted or . If is differentiable at ,
If and 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 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.
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 and 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, , are true if . 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. For example
The number e, also known as Euler's number, is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of natural logarithms. It is the limit of (1 + 1/n)n as n approaches infinity, an expression that arises in the computation of compound interest. It can also be calculated as the sum of the infinite series
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any finite sums, products and compositions of these, such as the trigonometric functions sine and cosine and their hyperbolic counterparts sinh and cosh, as well as derivatives and integrals of entire functions such as the error function. If an entire function has a root at , then , taking the limit value at , is an entire function. On the other hand, the natural logarithm, the reciprocal function, and the square root are all not entire functions, nor can they be continued analytically to an entire function.
In mathematics, the gamma function is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer n,
L'Hôpital's rule, also known as Bernoulli's rule, is a mathematical theorem that allows evaluating limits of indeterminate forms using derivatives. Application of the rule often converts an indeterminate form to an expression that can be easily evaluated by substitution. The rule is named after the 17th-century French mathematician Guillaume de l'Hôpital. Although the rule is often attributed to l'Hôpital, the theorem was first introduced to him in 1694 by the Swiss mathematician Johann Bernoulli.
The natural logarithm of a number is its logarithm to the base of the mathematical constant e, which is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of x is generally written as ln x, logex, or sometimes, if the base e is implicit, simply log x. Parentheses are sometimes added for clarity, giving ln(x), loge(x), or log(x). This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.
Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful. This page lists some of the most common antiderivatives.
In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input.
In mathematical analysis, an improper integral is an extension of the notion of a definite integral to cases that violate the usual assumptions for that kind of integral. In the context of Riemann integrals, this typically involves unboundedness, either of the set over which the integral is taken or of the integrand, or both. It may also involve bounded but not closed sets or bounded but not continuous functions. While an improper integral is typically written symbolically just like a standard definite integral, it actually represents a limit of a definite integral or a sum of such limits; thus improper integrals are said to converge or diverge. If a regular definite integral is worked out as if it is improper, the same answer will result.
In calculus and other branches of mathematical analysis, when the limit of the sum, difference, product, quotient or power of two functions is taken, it may often be possible to simply add, subtract, multiply, divide or exponentiate the corresponding limits of these two functions respectively. However, there are occasions where it is unclear what the sum, difference, product, quotient, or power of these two limits ought to be. For example, it is unclear what the following expressions ought to evaluate to:
In mathematics, tetration is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though and the left-exponent xb are common.
In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.
In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be stable if its distribution is stable. The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it.
In mathematics, there are several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet, one of which is the improper integral of the sinc function over the positive real line:
In mathematics, the Stieltjes constants are the numbers that occur in the Laurent series expansion of the Riemann zeta function:
In calculus, the Leibniz integral rule for differentiation under the integral sign states that for an integral of the form
In discrete calculus the indefinite sum operator, denoted by or , is the linear operator, inverse of the forward difference operator . It relates to the forward difference operator as the indefinite integral relates to the derivative. Thus