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Indeterminate form is a mathematical expression that can obtain any value depending on circumstances. In calculus, it is usually possible to compute the limit of the sum, difference, product, quotient or power of two functions by taking the corresponding combination of the separate limits of each respective function.
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
Other indeterminate forms, such as 1 ∞, 0 0, ∞ 0, 0 · ∞, and ∞ − ∞, can sometimes be evaluated using L'Hôpital's rule. We again indicate applications of L'Hopital's rule by = . For example, to evaluate a limit involving ∞ − ∞, convert the difference of two functions to a quotient:
An example of a limit where is not defined at is given below. Consider the function =. then f(1) is not defined (see Indeterminate form), yet as x moves arbitrarily close to 1, f(x) correspondingly approaches 2: [13]
This rule uses derivatives to find limits of indeterminate forms 0/0 or ±∞/∞, and only applies to such cases. Other indeterminate forms may be manipulated into this form. Given two functions f(x) and g(x), defined over an open interval I containing the desired limit point c, then if:
[22] Knuth (1992) contends more strongly that 0 0 "has to be 1"; he draws a distinction between the value 0 0, which should equal 1, and the limiting form 0 0 (an abbreviation for a limit of f(t) g(t) where f(t), g(t) → 0), which is an indeterminate form: "Both Cauchy and Libri were right, but Libri and his defenders did not understand why ...
Convolution. Cauchy product –is the discrete convolution of two sequences; Farey sequence – the sequence of completely reduced fractions between 0 and 1; Oscillation – is the behaviour of a sequence of real numbers or a real-valued function, which does not converge, but also does not diverge to +∞ or −∞; and is also a quantitative measure for that.
The expressions , (), and / (called indeterminate forms) are usually left undefined. These rules are modeled on the laws for infinite limits . However, in the context of probability or measure theory, 0 × ± ∞ {\displaystyle 0\times \pm \infty } is often defined as 0.