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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 provide sufficient information to determine the original limit, then it is said to assume an indeterminate form. More specifically, an indeterminate form is a mathematical expression involving 0, 1 and , obtained by applying the algebraic limit theorem in the process of attempting to determine a limit, which fails to restrict that limit to one specific value and thus does not yet determine the limit being sought. The term was originally introduced by Cauchy's student Moigno in the middle of the 19th century.
There are seven indeterminate forms which are typically considered in the literature:
The most common example of an indeterminate form occurs when determining the limit of 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 ". For example, as x approaches 0, the ratios , , and go to , 1, and 0 respectively. In each case, if the limits of the numerator and denominator are substituted, the resulting expression is , which is undefined. In a loose manner of speaking, can take on the values 0, 1, or , and it is easy to construct similar examples for which the limit is any particular value.
Not every undefined algebraic expression corresponds to an indeterminate form. For example, the expression 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.
Expressions that arise in other ways than by applying the algebraic limit theorem may assume the same form as one of the indeterminate forms. It is not appropriate, however, to call these expressions "indeterminate forms" outside the context of determining limits. The most common case is , which may, for example, arise from substituting for in the equation . This expression is undefined, as is division by zero in general. The other case is the expression . Whether this expression is left undefined, or is defined to equal , depends on the field of application and may vary between authors. For more, see the article Zero to the power of zero. Note that and other expressions involving infinity are not indeterminate forms.
As mentioned above,
This is enough to show that is an indeterminate form. Other examples with this indeterminate form include
Direct substitution of the number that approaches into any of these expressions shows that these are examples correspond to the indeterminate form , but these limits can assume many different values. Any desired value can be obtained for this indeterminate form as follows:
The value can also be obtained (in the sense of divergence to infinity):
The following limits illustrate that the expression 00 is an indeterminate form:
Thus, in general, knowing that and is not sufficient to evaluate the limit
If the functions f and g are analytic at c, and f is positive for x sufficiently close (but not equal) to c, then the limit of will be 1. Otherwise, use the transformation in the table below to evaluate the limit.
The expression is not commonly regarded as an indeterminate form, because there is not an infinite range of values that could approach. Specifically, if f approaches 1 and g approaches 0, then f and g may be chosen so that:
In each case the absolute value approaches , and so the quotient must diverge, in the sense of the extended real numbers (in the framework of the projectively extended real line, the limit is the unsigned infinity in all three cases). Similarly, any expression of the form with (including and ) is not an indeterminate form, since a quotient giving rise to such an expression will always diverge.
The expression is not an indeterminate form. The expression obtained from considering gives the limit 0, provided that remains nonnegative as approaches . The expression is similarly equivalent to ; if as approaches , the limit comes out as .
To see why, let where and By taking the natural logarithm of both sides and using we get that which means that
The adjective indeterminate 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.
For example, the expression 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.
When two variables and converge to zero at the same point and , they are called equivalent infinitesimal (equiv. ).
Moreover, if variables and are such that and , then:
Here is a brief proof:
Suppose there are two equivalent infinitesimals and .
L'Hôpital's rule is a general method for evaluating the indeterminate forms 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 expressions , , and so on, as these expressions are not indeterminate forms.) These derivatives will allow one to perform algebraic simplification and eventually evaluate the limit.
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. Note that this equation is valid (as long as the right-hand side is defined) because the natural logarithm (ln) is a continuous function; it is irrelevant how well-behaved f and g may (or may not) be as long as f is asymptotically positive.
Although L'Hôpital's rule applies to both and , one of these forms may be more useful than the other in a particular case (because of the possibility of algebraic simplification afterwards). One can change between these forms, if necessary, by transforming to .
The following table lists the most common indeterminate forms, and the transformations for applying l'Hôpital's rule.
|Indeterminate form||Conditions||Transformation to||Transformation to|