Indeterminate Form

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Indeterminate Form

 

in mathematics, an expression whose limit cannot be found by direct application of the usual limit theorems. Indeterminate forms may be of the following types:

Other indeterminate forms are (sin x)/x as x ←, 0, where

and [1 + (1/x)] x as x → ∞, where

(e = 2.71828 ... is the base of the natural logarithms). The seven types of indeterminate forms listed above are denoted as (1) 0/0, (2) ∞/∞, (3) 0 • ∞ ,(4) ∞ -∞ ,(5) 1∞, (6) 0°, (7) ∞ °. We observe that a given function can be an indeterminate form for some values of the independent variable and not for others [for example, the expression (sin x)/x as x x → π is not an indeterminate form]. Not every indeterminate form has a limit. The expression

as x → 0 does not tend to a limit:

but

Indeterminate Form

does not exist.

Finding the limit of an indeterminate form (if it exists) is often based on replacing a given function by a different function that has the same limit but is not an indeterminate form. This replacement is sometimes achieved by means of algebraic transformations. For example, if we divide the numerator and denominator in the expression (1 − x)/(1 − x2) by 1 − x, then we obtain 1/(1 + x).Therefore,

Indeterminate forms of types (1) and (2) can often be evaluated by means of l’Hôpital’s rule. L’Hópital’s rule asserts that for indeterminate forms of types (1) and (2) the equation

holds provided that f(x) and g(x) are differentiable in a neighborhood of x0 (x0 may be a finite point or a point at infinity) with the possible exception of x0, and provided that

exists. Using this rule, we find, for example, that

Indeterminate Form

It may be that [/”’(*)]/[ g’(x)] is again an indeterminate form of type (1) or (2). L’Hópital’s rule can then be applied once more (if the relevant conditions hold) and so on. This approach, however, is not always successful. For example, application of L’Hópital’s rule to the indeterminate form

[f(x) = ex + e-x and g(x) = ex - e-x] for x →0 pointless. It may also happen that

does not exist, whereas

of type (1) or (2) does. Thus

while

does not exist.

The expansion of functions in series is a powerful method for finding the limits of indeterminate forms. For example, since

we have

Indeterminate forms of types (3)-(7) can be reduced to either type (1) or (2). For example, as x ← π/2,

is an indeterminate form of type (4). Now

and as x − π/2 the latter indeterminate form is of type (1) and has the limit 0. An indeterminate form of type (3) is reduced to an indeterminate form of type (1) or (2) by the transformation f(x) g(x) = f(x)/h(x) or g(x)/k(x), where h (x) = g(x) and k(x) = f(x). Finally, if we denote by u(x) the logarithm of indeterminate forms of types (5), (6), and (7), that is, if we set u(x) = g(x) In f(x), then u(x) is an indeterminate form of type (3), which, as was indicated, can be reduced to an indeterminate form of type (1) or (2). Finally, consider {f (x)}g(x) = eu(x). If we can find the (presumably existing) limit of u(x) then we can also find the limit of the given indeterminate form. For example, for xx as x →0 we have

and, consequently,

REFERENCES

II’in, V. A. , and E. G. Pozniak. Osnovy matematicheskogo analiza, 3rd ed., part 1. Moscow, 1971.
Kudriavtsev, L. D. Matematicheskii analiz, 2nd ed., vol. 1. Moscow, 1973.
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The difference between English and Chinese in this respect is readily seen when (17) is translated into Chinese: the referent in question will most likely assume the form of an indeterminate expression (to be explained in Section 4) in a sentence position that does not make any clear indication or suggestion to the addressee regarding whether the expression is to be interpreted as identifiable or nonidentifiable.
Given that possibility, when the addressee cannot find a suitable identifiable referent satisfying the description of the indeterminate expression in the subject position, he will most likely read it in the same way as he interprets the indefinite subject in (108), treating it as a new referent making its first appearance in discourse, particularly when some other conditions are met (cf.
Finally, there are situations where the same indeterminate expression in an utterance can be interpreted either as an identifiable referent or a nonidentifiable referent.
For indeterminate expressions, there is a strong but seldom absolute correlation between the interpretation of identifiability or nonidentifiability and their occurrence in different positions in a sentence.
In most, but not all, instances, whether the indeterminate expressions are to be interpreted as identifiable or nonidentifiable is indicated, or suggested, by the position of the noun phrase in sentences.
When interpreted as of identifiable reference, indeterminate expressions in Chinese are used for referents which derive identifiability from the whole range of sources that is covered by the English definite article and demonstratives, as illustrated in the following sentences:
Indeterminate expressions on nonidentifiable reading also serve the major functions that are served by indefinite determiners like (yi)+ classifier exemplified in (72) through (74).
On the other hand, while statistically many more definite and indeterminate expressions occupy subject position, indefinite expressions serving as subject with dynamic predicate as in (71), repeated below as (108), are actually quite common in Chinese, so long as certain conditions are met (cf.
Other positions in sentences, like postverbal object in transitive sentences and oblique object, normally admit determinate expressions of either category, as well as indeterminate expressions.
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