Indeterminate form

A vague expression is in mathematics, a term whose appearance plays a special role in the study of limits. The term is to be distinguished from the undefined expression.

Problem representation

Since the division by zero is not defined, is the term 1: 0 not represent a number, when comparing with 1: x, where x is a very small (but positive ) number should be, the result is a very large value. For negative x the other hand, results in a corresponding negative value of large magnitude. It is therefore reasonable to introduce the symbol ∞, so that you can at least make the statement amount. Calculating with the extended to infinite elements of real numbers is possible with minor restrictions (please see Advanced real number ). Some terms such as 0: 0 is neither a number nor the symbol ∞, however, can also be assigned in such an extension.

Comparing the term 0: 0 with x: y, where both x and y amount small numbers, so their quotient as above may have a very large amount, but as well any other value. Even with the aid of so ∞ is from 0: 0 is not a suitable value is close, it is therefore an indefinite term.

Definition

Usually, the term " indefinite term" is used for:

It is exactly those expressions in which limit conclusions about the expression arise not only from the limit values ​​of the operands and even in the case of convergence different finite limits are possible.

Demarcation

Undefined expression does not mean the same as

• Undefined term: Numerous other terms are - not defined, about 1 - in the field of affine extended real numbers: 0 or. Conversely, it is quite common to define
• Discontinuity or non- liftable definition gap of the arithmetic operation: Otherwise would also 1: 0 are counted among the indefinite expressions.

No indefinite expressions (regardless of existence or finite ) limits of specific functions, such as

Although results from naive inserting here the indefinite expression 0: 0 and 0 · ∞. Through detailed examination with appropriate methods, such as the set of l'Hospital, the limit can be determined. It is

And not

Occurrence in consequence limits

Are and two sequences of real numbers, so you can see the consequences, and - if - define; as far as, for example, applies, too. If the output sequences converge in the affine extended real numbers, and about, then for the associated consequences and usually, one of the basic arithmetic operations and exponentiation respectively. However, if one of indefinite expressions listed above, the limiting behavior is undefined by. In fact, a (well partly ) arbitrary sequence are specified and then, be constructed, as the following list shows.

• 0: 0 Set and. Every now and then, or because.
• 0 · ∞ Set and. Every now and then, or because.
• ∞ - ∞ Set and. Then, and it is due, because, if, and if.
• ∞: ∞ It is assumed. Set and. Then, so, of course.
• 00, ∞ 0, ∞ 1 It is assumed. Set and determine consequences as above, with, and. With you and take care of the case, 00
• And with the case ∞ 0,
• And with the case 1 ∞

Occurrence in function limits

The methods used above for consequences can be easily generalized to functions. In this way can be found to any real number (or or) each indefinite expression, each real function (if necessary with the restriction ) has two real functions and for all, and. So Here, every finite or infinite value assume (possibly only non- negative) or even do not exist. In other words: From a knowledge of and no inference can be obtained when an indefinite expression. In contrast, applies to the basic arithmetic and exponentiation well, if it is a defined and not indefinite expression ( and defined in a punctured neighborhood of ever is ); if necessary, in this case the rules for computing are to be observed as they apply for the extended real numbers.

Fulfill the functions and the stronger requirements of Rule of L'Hospital, in particular with regard to differentiability, it can be with the help of possibly a statement about this limit edn make.

Survey

Let and be real functions and is a real number or one of the two symbolic values ​​or. It is assumed that the limits and either exist or that there is some divergence, which is symbolically expressed as a limit or. In most cases, note that then also the following limits with the specified values ​​exist (or certain divergence exists when there is right):

• ,
• ,
• ,
• .

Here, the calculation rules for, for, for, for, for, for, for, for, and corresponding Vorzeichenvarinaten were agreed.

However, the existence of the limit on the left, let alone its value does not show up in this simple manner from the limits of the operands, if the right one of the above indefinite expressions would result. In the following example functions are listed with the corresponding limits for which different exposure limits or divergence follows:

• 0: 0   with.   with.
• ∞: ∞   with.   with.
• 0 · ∞   with.   with.
• ∞ - ∞   with.   with.
• 1 ∞   with, unless.   with.
• 0 0   with, unless.
• ∞ 0   with, unless.   with, unless.

By mathematical transformations, the various types of indefinite expressions on the Type 1 can be traced back. With an indefinite expression of type 2, for example formed by the forming an expression of the type 1

Expressions of type 5-7 can be traced by taking the logarithm of the type 1.

The term ∞: 0 can in principle also not a complete statement of the boundary behavior, but this divergence can at least certainly not give a finite limit, but at the most specific, unlike the above-enumerated cases or after. As an example, consider with for and optionally

• : Certain divergence,
• : Certain divergence,
• : Left - and right-sided different specific divergence, ie a total indefinite divergence,
• : On one side there is even an indefinite divergence.

The term 00

A special role is played by the expression to defined itself well, namely as. For this purpose, it should be noted that the exponentiation, which is the calculation of the expression is defined as a first even only repeated multiplication, thus must be a non-negative integer. Independent of - - Then the empty product which is defined as 1: It applies to what follows, at least for mandatory. The empty product has no factors, and in this respect it does not matter what the value of not occurring factor, so that may also be seen. The definition also makes sense for other reasons. For example, there is, if both are non-negative integers, always exactly one - element set of pictures in one - element set. This applies only to the definition in the case.

The so as picture of by -defined operation of exponentiation can be in the real domain by also to the case continue and for non- negative by radicals initially on non-negative rational exponents and then by limit viewing on. The latter is by definition continuous in, however, exponentiation is as picture of not continuous at the point after a total of: for example, applies, however. From this discontinuity, the above-mentioned uncertainty results in the context of limits.