In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are often performed by replacing subexpressions by their limits; if the expression obtained after this substitution does not give enough information to determine the original limit, it is known as an indeterminate form. The indeterminate forms include 0⁰, 0/0, 1^∞, ∞ - ∞, ∞/∞, 0×∞, and ∞⁰. For other uses, see Calculus (disambiguation). ... Analysis has its beginnings in the rigorous formulation of calculus. ... An expression is a combination of numbers, operators, grouping symbols (such as brackets and parentheses) and/or free variables and bound variables arranged in a meaningful way which can be evaluated. ... A limit can be: Limit (mathematics), including: Limit of a function Limit of a sequence Limit point Net (topology) Limit (category theory) A constraint (mathematical, physical, economical, legal, etc. ...

Discussion

The most common example of an indeterminate form is 0/0. As x approaches 0, the ratios x²/x, x/x, and x/x³ go to 0, 1, and correspondingly. In each case, however, if the limits of the numerator and denominator are evaluated and plugged into the division operation, the resulting expression is 0/0. So (roughly speaking) 0/0 can be 0 or it can be and, in fact, it is possible to construct similar examples converging to any particular value. That is why the expression 0/0 is indeterminate.

More formally, the fact that the functions f and g both approach 0 as x approaches some limit point c is not enough information to evaluate the limit This article is about functions in mathematics. ... In mathematics, informally speaking, a limit point (or cluster point) of a set S in a topological space X is a point x in X that can be approximated by points of S other than x as well as one pleases. ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as...

That limit could converge to any number, or diverge to infinity, or might not exist, depending on what the functions f and g are.

Not every undefined algebraic expression is an indeterminate form. For example, the expression 1/0 is undefined as a real number but is not indeterminate. This is because any limit that gives rise to this form will diverge to infinity. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a divergent series is an infinite series that does not converge. ...

An expression representing an indeterminate form may sometimes be given a numerical value in settings other than the computation of limits. The expression 0⁰ is defined as 1 when it represents an empty product. In the theory of power series, it is also often treated as 1 by convention, to make certain formulas more concise. (See the section "Zero to the zero power" in the article on exponentiation.) In the context of measure theory, it is necessary to take to be 0. In mathematics, an empty product, or nullary product, is the result of multiplying no numbers. ... In mathematics, a series is often represented as the sum of a sequence of terms. ... In mathematics and in the sciences, a formula (plural: formulae, formulæ or formulas) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. ... “Exponent” redirects here. ... “Exponent” redirects here. ... In mathematics, a measure is a function that assigns a number, e. ...

Some examples and nonexamples

The form 0/0

The indeterminate form 0/0 is particularly common in calculus because it often arises in the evaluation of derivatives using their limit definition. For other uses, see Calculus (disambiguation). ... This article is about derivatives and differentiation in mathematical calculus. ...

As mentioned above,

while

This is enough to show that 0/0 is an indeterminate form. Other examples with this indeterminate form include

and

Direct substitution of the number that x approaches into any of these expressions leads to the indeterminate form 0/0, but the limits take many different values. In fact, any desired value A can be obtained for this indeterminate form as follows:

Furthermore, the value infinity can also be obtained (in the sense of divergence to infinity):

The form 0⁰

The indeterminate form 0⁰ has been discussed since at least 1834.^[1] The following examples illustrate that the form is indeterminate: Year 1834 (MDCCCXXXIV) was a common year starting on Wednesday (link will display the full calendar) of the Gregorian Calendar (or a common year starting on Monday of the 12-day slower Julian calendar). ...

Thus, in general, knowing that and is not sufficient to calculate the limit

However, if the functions f and g are additionally assumed to be meromorphic on a neighbourhood of 0 in the complex plane then the limit of f ^g as z approaches 0 will always be 1. A meromorphic function is a function that is holomorphic on an open subset of the complex number plane C (or on some other connected Riemann surface) except at points in a set of isolated poles, which are certain well-behaved singularities. ... This is a glossary of some terms used in the branch of mathematics known as topology. ...

There are settings where 0⁰ is taken to be defined even though it is an indeterminate form, as discussed in the article on exponentiation. “Exponent” redirects here. ...

Undefined forms that are not indeterminate

The expression 1/0 is not an indeterminate form because there is no range of distinct values that f/g could approach. Specifically, if f approaches 1 and g approaches 0, then |f/g| must diverge to infinity. Notice that although f and g may be chosen (on an appropriate domain) so that f/g approaches either positive or negative infinity (in the sense of the extended real numbers), this variation does not create an indeterminate form (from one point of view, because they both diverge; from another point of view, because all infinities are equivalent in the real projective line). The extended real number line is obtained from the real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers). ... In mathematics, the projective line is a fundamental example of an algebraic curve. ...

Similarly, the expressions and are not indeterminate because any limit that gives rise to one of these forms will converge to 0 or diverge to infinity, respectively.

Evaluating indeterminate forms

The indeterminate nature of a limit's form does not imply that the limit does not exist, as many of the examples above show. In many cases, algebraic elimination, L'Hôpital's rule, or other methods can be used to manipulate the expression so that the limit can be evaluated. In calculus, lHôpitals rule (also spelled lHospital) uses derivatives to help compute limits with indeterminate forms. ...

For example, the expression x²/x can be simplified to x at any point other than x = 0. Thus, the limit of this expression as x approaches 0 (which depends only on points near 0, not at x = 0 itself) is the limit of x, which is 0. Most of the other examples above can also be evaluated using algebraic simplification.

L'Hôpital's rule is a general method for evaluating the indeterminate forms 0/0 and ∞/∞. This rule states that (under appropriate conditions)

where f' and g' are the derivatives of f and g. (Note that this rule does not apply to forms like 0/∞, 1/0, 2/3, and so on; but these forms are not indeterminate either.) With luck, these derivatives will allow one to perform algebraic simplification and eventually evaluate the limit. In mathematics, the derivative of a function is one of the two central concepts of calculus. ...

L'Hôpital's rule can also be applied to other indeterminate forms, using first an appropriate algebraic transformation. For example, to evaluate the form 0⁰:

The right-hand side is of the form ∞/∞, so L'Hôpital's rule applies to it. Notice that this equation is valid (as long as the right-hand side is defined) because the natural logarithm (ln) is a continuous function; it's irrelevant how well-behaved f and g may (or may not) be as long as f is asymptotically positive. The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...

Although L'Hôpital's rule applies both to 0/0 and to ∞/∞, one of these may be better than the other in a particular case (because of the possibilities for algebraic simplification afterwards). You can change between these forms, if necessary, by transforming f/g to (1/g)/(1/f).

List of indeterminate forms

The following table lists the indeterminate forms for the standard arithmetic operations and the transformations for applying l'Hôpital's rule.

See also

• Defined and undefined
• Division by zero
• Extended real number line

In mathematics, defined and undefined are used to explain whether or not expressions have meaningful, sensible, and unambiguous values. ... For the album by Hux Flux, see Division by Zero (album). ... The extended real number line is obtained from the real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers). ...

References

1. ^ www.faqs.org