Indeterminate form

In calculus and other branches of mathematical analysis, limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits; if the expression obtained after this substitution does not give enough information to determine the original limit, it is said to take on an indeterminate form. The term was originally introduced by Cauchy's student Moigno in the middle of the 19th century.

The indeterminate forms typically considered in the literature are denoted 0/0, ∞/∞, 0 × ∞, ∞ − ∞, 0⁰, 1^∞ and ∞⁰.

The most common example of an indeterminate form occurs as the ratio of two functions, in which both of these functions tend to zero in the limit, and is referred to as "the indeterminate form 0/0". As x approaches 0, the ratios x/x³, x/x, and x²/x go to ∞, 1, and 0 respectively. In each case, if the limits of the numerator and denominator are substituted, the resulting expression is 0/0, which is undefined. So, in a manner of speaking, 0/0 can take on the values 0, 1, or ∞, and it is possible to construct similar examples for which the limit is any particular value.

More formally, the fact that the functions f(x) and g(x) both approach 0 as x approaches some limit point c is not enough information to evaluate the limit

Not every undefined algebraic expression corresponds to an indeterminate form. For example, the expression 1/0 is undefined as a real number but does not correspond to an indeterminate form, because any limit that gives rise to this form will diverge to infinity.

An indeterminate form expression may have a value in some contexts. For example, if κ is an infinite cardinal number, then expressions 0^κ, 0⁰, 1^κ and κ⁰ are well defined in the context of cardinal arithmetic. See also § Zero to the power of zero. Note that zero to the power infinity is not an indeterminate form.

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